Perk requirement for spells?

The novice, apprentice, adept, and expert level spells are gotten from purchasing them (or in some cases, you can get them for free as loot or a reward).

The master level spells are acquirable generally after you complete your school-specific quests as a member of the College of Winterhold.

Perks don't add spells to your repertoire: they simply enhance the ones you already have.

I don't believe you have to be a certain level to cast any spell or have the perk. You just have to have enough magicka to cast it. I know this because I was casting the storm atronach spell very early. The perk just halves the amount of magicka it takes to cast spells of that level.