Personally I would table the Eldritch Knight idea.
While fun, they take quite a bit of time to be an effective melee combatant and before they get in a couple EK levels they are really just wizards that can fight a bit when they have backup.
For a dedicated front line combatant you probably need something sturdy and consistently dependable. It sounds like your flavor is a typical dwarf that self enchants with magic.
Core does make this more difficult; I would recommend a bloodrager or Magus if it weren't for that. Maybe a Paladin? You get spells you can buff yourself with (I believe including Divine Favor, which is one of the best buff spells). Perhaps a cleric? Very magic heavy, yet still durable and left over heals are certainly handy.
Regardless I wish you good luck with your dwarf-nanigans.
Uh, illusion includes mirror image, your best personal defence spell. Color spray could be a life-saver for you early on. Don't dump it. Necromancy is more dumpable, false life just isn't as important.
If I *had* to go core Dwarf Eldritch Knight, I'd go Barbarian 1 with some Extra Rage. You don't need to be able to cast spells and fight at the same time; you Rage when you're done casting spells and getting into melee. The last thing a melee character wants to do is pick fights that they're really bad at finishing.
Alternatively, core Eldritch Knight works very nicely with Weapon Finesse, Power Attack and elven curved blade. Dwarves have no more STR bonus than DEX bonus, which is to say they get neither. Transmuter works well with Finesse and curved blade; you get a DEX bonus item for attack, and put the Transmuter bonus into STR. You can still even take Barbarian 1 for the bonus damage; the AC penalty is then offset by higher DEX.
Note that the Transmutation ability score bonus is an Enhancement. It won't stack with items or most spells. So beyond very early levels, it's not very effective to be using it on STR. An Eldritch Knight with Transmutation eventually gets a +2. Personally I would redo those ability scores to something that suits a Wizard/melee; you're talking about cracking skulls and "offense is the best defense", but you've got weak attack and damage as well as weak defense.
Ditching Illusion means no Mirror Image, no Displacement, no Greater Invisibility... it's not just ditching "sneaking". You *can* go that way, but creating a melee Wizard while ditching really good Wizard defensive spells doesn't sound like a great plan. Mirror Image is arguably the best reason to go Core Eldritch Knight.
If you want a spells-and-melee Core Dwarf, a fighting/casting Cleric would be an awful lot easier to pull off. Just saying.
If you are looking to play as a frontliner I would also consider the symbol striker bloodrager archetype, it has that runemage-warrior theme that you are looking for. At level 6 you can literally inscribe runes on your weapon. I know that you are a dwarf (-2 cha) but you would be a bloodrager your cha doesn't need to be that high.
There's also the sigilus magus archetype that has a similar concept if you want to be int based.
In this way you can be an effective frontliner from the start instead of having to wait until you have a couple of levels in EK.
Eldritch Knight basically "breaks even" with full BAB, since they lose 3 BAB getting to EK levels but gain 2 attack from Heroism (which lasts long enough to be pretty much "always on") and they usually pay 1 less Power Attack penalty. However, martials typically get more than just full BAB...
The biggest question is what kind of buffing you can easily use. Heroism should usually be fine - with a cheap Extend Metamagic Rod, it lasts hours. If your GM/party makes it fairly easy to do things like cast Mirror Image when you know a fight is coming soon-ish or cast Haste as you move into battle, great. If you have to start tough fights without anything set up yet, you're going to need to worry a lot more.
Eldritch Knights are tricky to do on 20 PB, and dwarf complicates this since its racial bonuses are in your tertiary ability scores. EK's are rather MAD, and the physical ability scores you've assigned are a bit too low. Ideally I'd want to have 18 Str and 16 Dex (after racial and transmutation bonus) on a melee Eldritch Knight. This is doable in a pinch, but you'd need to drop constitution down to 12, intelligence to 13, and charisma down to 5. That might be a bit too steep for you, but I would definitely recommend trying to find some more points for your Str and Dex since they are so important for an EK.
School: right choice; transmutation is amazingly good for eldritch knights since you're so starved for ability scores, and every extra +1 helps. The change to an Evocation/Necromancy opposition is perfect.
Arcane Bond: keep in mind there's a 1 week period in which you'll have no arcane bond if you're switching items. In many adventures you won't have the opportunity to make the swap, and being stuck with a bonded weapon is catastrophically bad. It also costs you 200 gp to make the swap, so long-run this isn't saving you very much. Personally I don't think it's worth it, even for something as tempting as "free masterwork weapon at 1st". Secondly, go with a two-handed weapon. Preferably one with reach. You need a free hand to cast spells anyways, and it's a free action to change your grip.
1) Given that we 100% lack a melee character besides me, I may be able to convince the GM to let me start my first level off as a Fighter instead of a Wizard. This would give me better starting HP, but I'd have to get rid of whatever armor I start with once I hit level 2 and start casting spells.
That would be preferable; the extra starting HP would be very helpful, and once you reach 2nd it's basically just free HP.
2) If I start Fighter, I won't feel obligated to take Toughness as a bonus feat. That's always nice to not have to take. I can get Weapon Focus (Dwarven Waraxe) instead, or Power Attack, or both.
Power Attack isn't quite so great for Eldritch Knights. Due to your class levels in wizard, your attack bonus is a bit more iffy so you'll probably only use it when you have your buffs up. It's still not terrible, but it's definitely something further down on the list of priorities. Weapon Focus, on the other hand, is a great pick since it shores up one of your weak points. Combat Reflexes is another nice one; remember that you can self-buff with enlarge person to give yourself greater reach, and any character making use of reach enjoys combat reflexes.
3) As the game will be Core Only, I'm assuming that any non-Core item will be unavailable to me. Therefore, I can't hope for Celestial Full Plate or the like. I can, however, hope for Mithral, so with Mithral and Arcane Armor Training a lot of things become possible.
Keep in mind that going the arcane armor training route means you cannot use quicken spell. This is extremely painful at higher levels, and at a certain point you will be ditching armor. Depending on where you campaign stops this may not be an issue, but do keep in mind that quickened spell is a matter of when and not if.
4) I may be able to ask to adjust my stats.
If you can, definitely do so. Every point of Str and Dex you can gain will be huge.
There are problems for an EK with a shield. Heavy Shield and weapon blocks spellcasting, so you're stuck needing to draw/sheathe your weapon to go between fighting and casting. Quick Draw helps. Light shield or buckler doesn't block casting since you can hold the weapon in your shield hand for a moment.
But the biggest concern with a shield is that your damage is probably going to be pretty weak if you aren't serious about improving it. If you use a shield, you're going to want to be very aggressive - Good STR, Arcane Strike, Weapon Specialization at 9, Power Attack.
For a shield-using Dwarf EK, I'd do something like 16STR, 14DEX, 12/14CON, 15INT, 8/10WIS, 8\6CHA, put points into STR at 4 and 8, and get a +4STR belt. You've got a bunch of shield AC to shore up defense, so you can focus on being quite good at beating things with an axe. With Heroism and Haste, solid one-handed damage can do just fine.
Edit: Armor Training is a big question-mark. Eventually getting Quicken Spell can be very useful, but it eats high-level spell slots and comes quite late for a core EK - you need to be level 13 even just to burn one precious level 6 spell slot for a Quickened Scorching Ray.
Come to think of it, you could use a shield and Two-Weapon Fighting. 15DEX and the +2 you can get from Transmuter gives enough DEX for proper dual wield. Go 16STR (+6 from levels and belt), 15DEX (+2 from Transmuter), 12/14CON, 15INT, 7/9WIS, 8\6CHA. Then you can wield a war axe and light shield with pretty good power, and you have good shield and DEX AC.
Responding to the original post: Looks great I don't even think you need to change your oposition schools.
Grab yourself a reach weapon and combat reflexes. You cast spells on your turn and make attacks as enemies close to get you. With the stats you have you'll make good use of Enlarge Person.
There's a lot of advice about losing spells, but you have a sorcerer and a bard in your party, it doesn't matter as much as people think. The only spell i can think of that really matters is Mirror Image, and if you decide you need it youbcan cast spells from your opposition school.
I'd look at your role in the party, the Ranger is going to be your main damage dealer, the bard will amplify that while covering a lot of utility. From there it's up to you and the sorcerer to cover debuffing the enemies (and provide the utility that the bard can't). Really that's not going to be too hard.
If I were doing this I'd take Toughness and Combat Reflexes at level 1, Power Attack at 3. Then you build toward Arcane Armour Training/Mastery so you can end up with Mithral Full Plate and only 5% arcane spell failure. People will tell you that any spell failure is too much, but as a 7th level caster you'll have 18 spells per day (including bonus spells from high INT and specialization) plus cantrips. That's ~1 spell per day that fails from armour - totally not game-breaking. In fact you'll have to prepare less spells because your armour is already high (no mage armour for you). The real cost is the loss of your swift action casting, but that won't matter till higher levels, and even then you'll have a 75% success rate (that's a 56.25% chance that if you cast 2 spells in a round they both go through). It may not be the most optimal, but a Dwarf in full plate wielding a Ranseur who can reshape the battlefield with a thought is a pretty memorable character.
For a core-only dwarf wizard/eldritch knight, I'd recommend one of two routes:
1) Melee focus - barbarian 2 (to pick up Moment of Clarity)/wizard 5/eldritch knight X is the quickest route; however, barbarian 2/wizard 2/bard 1/dragon disciple 4 (advance wizard spellcasting)/eldritch knight X may be a better combatant.
2) Ranged focus - (fighter or ranger) 1/wizard 6/eldritch knight 2/arcane archer 3/eldritch knight +8.
As far as Sorcerer and Dragon Disciple goes, it's not as bad for a Dwarf as it may at first seem. Dragon Disciple grants +4 STR and +3 AC in 4 levels, which is great for taking pressure off physical ability score needs.
In ability score terms, starting with 16STR, 12DEX, 12/14CON, 10INT, 7/9WIS, 16\14CHA, and putting the points at 4 and 8 into CHA, you would still end up with 20STR, 16 in your casting stat, and the same AC as if you had 18DEX by Dragon Disciple 4. At that point you can go Eldritch Knight. You would be very hard pressed to get a Wizard/EK up to 20STR and 16INT that easily.
I think this is where I'm going to go with my character idea:
Level 1 = Fighter, 2-6 = Wizard, 7-16= Eldritch Knight, 17-20 = Wizard
OR
Level 1 = Barb, 2 = Wizard, 3 = Barb, 4-7 = Wizard, 8-17 = Eldritch Knight, 18-20 = Wizard
Whichever is the case, my stats will be:
20 Point Buy, Dwarf
STR 16
DEX 14
CON 15 (13+2)
INT 15
WIS 9 (7+2)
CHA 7 (9-2)
I'd go with your first stat array, with 15 STR and 16 CON - it costs less because of your racial modifiers and you don't have to dump WIS so hard. Your transmuter point goes into STR, you bump INT at level 4, at level 6 (fighter 1, wizard 5) your transmuter bonus goes up to 2, so at 8 you bump STR. You now have 18 STR by level 8.
It's worth noting that the transmuter bonus doesn't stack with a belt of giant strength/etc (they're enhancement bonuses). When you get a belt you can put the +2 into CON (18 CON makes even a wizard tough) or DEX (more armour, more initiative, more AoO's), essentially letting you wear 2 "belts".
I also don't save a point for my primary stat for level 20. Most games don't get there, and those that do often end there, so you might as well get it maxed at 16 to actually get the benefit out of it.
Arcane Strike will eventually get you +4 damage, so it's definitely not a bad feat, but remember that your EK levels count as fighter levels, so you can eventually get Weapon Specialization for +2 damage that doesn't eat your swift action.
It looks pretty good; I gotta say though, you're pretty close to being able to do the Transmutation School thing where 15DEX plus the +2DEX you get from Transmutation at Wizard 5 eventually lets you go Two-Weapon Fighting, Improved Two-Weapon Fighting and probably Double Slice. I crunch numbers when bored, and TWF war axe and shield ain't bad at all. Thinking about it, Core Eldritch Knight really has a whole lot of feats to use (since feat options are really limited with core), and with feats you can both gain a shield bonus and get two decent bonus attacks with a spiked light shield with feats to spare.
Now, if you decide to go two-hander and rely on Arcane Armor Training for AC, then swing away. A reach polearm would be a great possibility; most reach polearms are just big long "axes" anyhow, and reach polearms mean bonus attacks of opportunity even if you're casting spells on your turn. Move up almost to frontline to tempt enemies, cast spell, whack enemies who approach you. The Shield spell is pretty good Shield AC without a shield, but of course that comes back to the question of what you can have running when combat starts.
As far as abilities go, I'll note that +1 hp per level = Toughness. +1 attack and +2-3 damage because you have more STR is equal to two feats at least.