Necromancy Spells: Command Undead, False Life, Enervation, Disrupt Undead, Mathematical Curse, Ray of Exhaustion, and Possession
Check out Proffessor Q's guide to wizardry for a big list of evaluated spells.
Last question. No
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That's a surprisingly lean number of spells for a 12th level Wizard to know. Assuming he started with an 18 Int score (if he didn't, why is he a Wizard), he would have all 0-level spells minus his opposition school spells and 7 1st level spells (3+Int bonus). At 2nd level he'd get two more for at bare minimum, 9 1st level spells.
Generally speaking, a Wizard will always take the highest level of spell he can cast at a given level for his two free spells per level, giving him at minimum 4 spells of each level. If he's been adventuring from level 1-whenever he'll have even more spells than that either by capturing enemy spellcaster spellbooks, finding spellbooks or scrolls as treasure, or spending gold to purchase more spells from other Wizards.
As for opposition schools, unfortunately no, the elemental schools aren't valid opposition schools unless your specialized school happens to be an elemental school. I've played a LOT of necromancers over the years and since Pathfinder opened up the ability to choose your opposition schools, I prefer to drop Enchantment and Divination. Enchantment is a crap school unless you are expressly specializing in it and Divination, for the most part, is a school that works best out of combat or from a scroll. Never drop Illusion as it has most of the best defensive spells, nor Conjuration as it has most of the non-SR spells.
Onto spell choice. I'll give you my proposed list based off that painfully limited number you're being allowed.
0-level:
Prestidigitation - One of the best cantrips, period.
Acid Splash - a good non-SR backup attack spell. Hold an Acid Flask while you cast it for a +1 damage boost.
Disrupt Undead - This is a solid anti-undead attack spell, and as a Necromancer you should have it or you aren't representing the profession.
Dancing Lights - As a Dhampir, you don't need a light source, but most likely your party members do. Unlike Light, this has the added benefit of allowing you to light up areas at a distance that are well outside even your darkvision range.
Mending - Your gear will break. This deals with it.
Detect Magic - Since you aren't getting the minimum number of 0-level spells that any other Wizard should have, I'd argue it's only fair you can take this without the opposition school restriction (assuming you took my advice regarding Enchantment & Divination).
Read magic - Same argument as for Detect Magic.
1st level:
Chill Touch - On its own, it's a risky spell to use, however it pairs VERY well with Spectral Hand and can send Undead fleeing which can be good for battlefield control. Plus, it can be used up to 12 times per casting as a 12th level caster. Good to put in your Specialist School spell slot.
Protection From Evil - A good defensive spell and valuable for its ability to ward off mental control from evil casters.
Mage Armor - Always have this on.
Magic Missile - You're go to 1st level attack spell.
Enlarge person/Reduce person - Which one you take depends on if you want to enhance a melee fighter or a ranged fighter. Enlarge for melee, reduce for ranged.
Comprehend Languages - Even as a Wizard you won't have all the languages and when you come across something weird, you almost certainly need to be able to read it or suffer.
Endure Elements - When extreme environments come up, you and the party will be happy you have this. If you can start with a wand of this instead, then I would swap it out for Obscuring Mist or Grease.
Bonus: Snowball - This is a fantastic ranged touch spell that is also non-SR. However it comes from an Adventure Path/Player Companion, so your GM may restrict your access to it. If you can take it, trade out Endure Elements or Enlarge Person for it.
2nd level:
Glitterdust - Easily the single most useful 2nd level spells. Shuts down invisibility as an AoE, is non-SR and can blind targets.
False Life - Always have this up. it's 13-22 temporary hitpoints for 12 hours at your level. Those can make the difference.
Spectral Hand - Touch spells tend to be very powerful because they're meant to be balanced by the fact that Wizards are weak in melee. This circumvents the single drawback of touch spells.
Blindness/Deafness - Use this to shut down enemy spellcasters. Fort saves are generally their weakest save and you can't cast a spell if you can't target and you can't target if you can't see.
Mirror Image - A fantastic defensive spell. As a Wizard you will never win the AC arms race. Instead, go a different route and benefit from miss chances.
Eagle's Splendor/Fox's Cunning - If you want to increase your ability to use Power Over Undead, take Eagle's Splendor. If you want to up your spell DCs, Fox's Cunning.
Minor Image - This one is GM/your imagination dependent, but can be used to great affect to distract enemies or lead them into traps.
Note: Don't waste a spellbook page for Command Undead. It's a crap spell that targets a single undead creature. Use Power Over Undead instead since it doesn't waste a spell slot, targets multiple undead and is easier to scale the DC up using magic items/Eagle's Splendor/a good Charisma score.
3rd level:
Dispel Magic - You will need this eventually.
Fireball - You will need this often.
Spiked Pit - The Pit line of spells are all awesome, but Spiked Pit adds some extra oomph from increased depth (so more falling damage) and spike damage. Sweep people up with Aqueous Orb and dump them in the pit. Drop a Cloudkill into it for added fun.
Aqueous Orb - Pairs well with Spiked Pit or to mop up stragglers who avoided Black Tentacles.
Fly - You will always need this. Don't forget to max out your Fly skill.
Haste - Your party will love you. Plus it makes it easier for you to run away.
Vampiric Touch/Gentle Repose - Depends on if you want to use Spectral Hand or make pretty undead/keep fallen comrades fresh longer.
4th level:
Protection From Energy; Communal - Worth a 4th level slot when you know you're facing a dragon. You're party will thank you.
Dimensional Anchor - Prevent a****ole spellcasters or demons from bugging out early.
Black Tentacles - Wonderful for battlefield control. Beware falling victim to your own tentacles though.
Dimension Door - Be an a****ole spellcaster. ^_^
Animate Dead - Let's face it. This is why we play necromancers.
Mnemonic Enhancer - This spell is far better than people give it credit for. Use the version that allows you to prep more spells, like so. If you have one day to prepare, use that day to fill all your 4th level and higher spells with Mnemonic Enhancer but wait to cast them until the end of the day before you sleep. Cast them for the added spell slots and fill them with Fireballs/Magic Missiles, Snowballs, whatever. Sleep. Now fill your normal spell slots with whatever. The Mnemonic Enhancer slots last 24 hours, so you'll get most of an adventuring day out of them without touching any of your real spell slots.
Bonus: Calcific Touch - This one you should take if you're going for a Spectral Hand build. It is a possible Slow effect and a guaranteed 1d4 Dex damage. If you can break their SR and roll max damage (or just maximize the spell), you can petrify an Ancient or older Dragon in two rounds. Enervation is also a good choice.
5th level:
Break Enchantment - At 12th level, this will come up.
Cloudkill - So. Much. Fun.
Life Bubble - Give it to your party, drop a Cloudkill all around your party in the middle of a fight and enjoy the show.
Teleport - This one sells itself.
Repair Undead, Mass - For keeping your animated legions up in tip-top condition longer.
Permanency - Ohhh, the possibilities. Becomes even more awesome if your GM allows spells beyond those suggested in the Core book.
6th level:
Disintegrate - It's too much fun not to use.
Contingency - Super useful for survival purposes.
Undeath to Death/Circle of Death - Depends on how much Undead fightin' you expect to do. Either one is very Necromancery
Antimagic Field - This one is risky to use, but can seriously turn the tide when fighting a spellcaster. Step one. Exhaust your normal spells. Step two. Cast Antimagic Field. Step three. Go toe to toe with the baddie. Step four. Have your party members keep the fight next to your unconscious body/corpse since the spell doesn't end until you dismiss it or the duration expires.
Wall of Iron - Battlefield control/slapstick attack spell.