no, because עוגת is construct state and can't stand alone. You can use עוגת השוקולד, עוגת התפוזים etc.
It's correct, but the spelling is wrong - it should be אל תיגע בעוגה
I don't have a definitive answer because the rules of writing without nikkud are rather flexible, and it's not really wrong to write תגע (just less common and clear).
But generally - in pa'al, when the 'נ disappears you'd add 'י, yes.
Is a י always added when a נ or other letter disappears in the future tense?