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Flexible school only gives you what it say nothing less, nothing more. You are not gaining an extra school all you gain are the power from the school. You do not gain extra spells slots and you still have to use your school spell slots for spells from your actual school. You cannot use those slots for spells from the flexible school. You do not need to choose additional oppositions schools for the same reason.

Awesome, thank you!

I'm looking at the Archmage ability called Flexible School:
"Select one wizard arcane school that is not your arcane school. You gain powers from that arcane school, treating your tier as your wizard level for the purposes of these powers. Once you have chosen the arcane school, it cannot be changed. You cannot select an arcane school that is one of your opposition schools.

You must have the arcane school class feature to select this ability. You can choose this ability up to three times, each time selecting another arcane school other than your own."

However, the arcane school feature in the wizard class mentions that for each school, they take a number of powers, and additional spell slots:

"Each arcane school gives the wizard a number of school powers. In addition, specialist wizards receive an additional spell slot of each spell level he can cast, from 1st on up. Each day, a wizard can prepare a spell from his specialty school in that slot. This spell must be in the wizard’s spellbook. A wizard can select a spell modified by a metamagic feat to prepare in his school slot, but it uses up a higher-level spell slot. Wizards with the universalist school do not receive a school slot."

Does this mean that archmages can get 2 spell slots for each spellcasting level, and does that mean that they must choose two more opposition schools?

My gut feeling says no, but I wanted to make sure.

At what level? To a single target, or to an area, or what?

Ideally, up to level 20, and to an area, but if single target deals a lot more damage, then go for it. Hopefully you'd be working alone, blasting off an evocation spell like Hyper Fireball or Ray of Heck your HP, but if a summon can just dwarf anything the wizard can do by themselves, then be sure to mention it.

As said above, I'm looking for the most damage a wizard can do within a single round. Metamagics, mythic spells, etc, how can one optimize to be an incredible blaster caster?
I'm rather new to Pathfinder, so I'd like to focus on wizard only, as even though they might optimize for blasting, their varied spells allow them to fill out many utility rolls, and I'm aware that other spellcasters have options to dwarf the wizard's damage output, so giving mentions to them would be really helpful.
I'm also interested in how a player might build towards such a goal?
My first impression is to become an Evoker wizard, but I imagine with the Mythic Heroes rules, you can do a lot more with it.

I would gravitate toward the Spell Sage for incredible power and versatility at higher levels. It will give you one shot of high damage early on, but you may need more steady damage, from the sound of things. For now, is a really big damage bump once a day workable, or do you need less but steady? Fair warning, at very low levels, wizards are simply not the big damage class. If they expect you to pull major DPS duty, they're barking up the wrong tree until a you get a few levels under your belt. Even the classic wizard runs out of slots quickly.

Word to the wise - for those utility spells, scrolls are your best friend.

Steady damage may be more necessary, and I've got plenty of scrolls for situational but useful spells.

Do you have any idea what others are planning? If there's three people planning to make melee beasts, a wizard is unlikely to have a lot of moments to shine as a shapeshifter.

The other three players are a sort of gish magus build, a hit sponge fighter, and a psychic class that focuses on enchantment, with some tricks to get past the immunities.

I am talking about three things: The classic wizard, the Spell Sage archetype, and the Shapeshifting Mastery Path from the Archmage. I want to know which would be the best 'path' to go down for a campaign with Mythic Heroes.
I'm lucky enough to be part of one, although I've been having problems deciding on which of three paths I may go down during a potentially 2 year campaign.

The classic wizard is the first option. Can't really go wrong, you have a solution for most, if not all problems, and can pull perfect solutions out of a hat with Wild Arcana.

The Spell Sage suffers early on from more limited spell slots and resources (although it's more bearable with Wild Arcana), but makes up for it down the line with unparalleled versatility, having access to the cleric, bard, and druid spells (useful especially in a party that potentially lacks divine casters).

Lastly, the Shapeshifting Mastery path allows the wizard to fully use polymorph spells at their full potential, with full BAB and multiple natural attacks and such, but requires an investment in more 'fighter based' things like stats and feats to reach that potential.

I've been having trouble deciding which of these three options to utilize.
Should I build towards having a backup that can be a force in combat when it requires such a heavy investment in less wizardy things?
Would the versatility of spells at my disposal make up for the limited resources I'll have, especially early on?
Or should I specialize on the school I'll be using the most and focus only on my strengths?