I put אף פעם before אני and the sentence has been rejected. Why does it have to be after?
It's acceptable, but not very natural. The most natural is "אני אף פעם לא הולך לגני חיות", second best is "אני לא הולך אף פעם לגני חיות" in my opinion.
אני לא הולך לגני חיות אף פעם
it isn't a thing of logic? אפ פעם is a negative thing... and לא also is a negative thing!.... double negative thing is an affirmation! ....
Double negatives don't make a positive in all languages. But if it helps, literally "אף פעם" means "even once", so "אני אף פעם לא הולך" literally means "I don't go even once".
Why is it with -ל instead of אל?
Because the word "אל" means "towards". For most purposes you should use "ל-"
Is there something wrong with using לעולם instead of אף פעם?
Yes. לעולם is used only in the future tense. אף פעם is used in the present tense and מעולם in the past tense.