Do higher level spells become accessible when you have a certain level of the skill or do they become available only after acquiring the perk?
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.
You will need to be a certain level to find spells,otherwise they won't exist eg.if you aren't level 25 or higher you can't find telkinisis in the labyrinthian.Perks only improve the efficiency of spells,making them cost less magicka and do more damage