If I make a character who takes all of the 5/5 sneak perks and the whole left side of the tree as well as the illusion quiet casting perk would I be able to run around casting destructions spells without enemies finding me?

Lol probably. I use Skyrim Redone and can't remember how most of the vanilla perks/skills operate, but do you get sneak attack bonuses when using spells?

Not extra damage bonuses. But kind of a hidden bonus- if you're not detected your magicka regenerates at the normal rate. If you're seen, magicka regenerates at the slower combat rate.

Yes. I'm doing that now with my mage. Casting invisibility, running into a room full of enemies and setting off firestorm before anyone knows I'm around never gets old.

Don't bother getting any sneak perks. Boots with the muffle enchantment and fortify sneak are all you need.

With that strategy, you will get caught, right? Can you cast a spell then run to a different corner, cast another, and run to a different area without getting caught? Enemies will search where the spell came from. Also, the perk in the sneak tree where running does not affect detection might be helpful.