Received a PM asking me to respond.

Present: I GO to school in NY

Yes, that's right.

Present Perfect: I will be GOING to school in NY

This is actually future continuous or progressive (not tested on the exam usually). Present perfect is constructed with "have / has" and the past participle. For regular verbs, the past participle is typically an -ed word. "to go" has the irregular past participle "gone."

Present Perfect: I have gone to school in NY

Past: I WENT to school in NY

Yes.

Perfect Past: If I HAD GONE to school in NY, then I WOULD HAVE WENT to class early everyday

Not quite. "gone" is the past participle. "went" is how we conjugate the verb in past tense. past perfect is constructed using "had" plus the past participle: had gone.

these are all simple past tense: I went, you went, he went, we went, they went

these are all past perfect: I had gone, you had gone, he had gone, we had gone, they had gone

By the time you arrived, we had already gone to bed.

The sentence you offer above has the additional complexity of being an "if/then" statement that does NOT reflect reality - that is, it's a hypothetical. In that case, you'd say:
If I had gone to school in NY, then I would have gone to class early every day.

The "would have gone" expresses a conditional (and hypothetical) statement - this only would have happened IF something else had happened first. This is also not that common to see on the GMAT.

Future: I WILL GO to school in NY once I get accepeted to NYU

yes.

Future Past: I WILL HAVE to do my homework by tomorrow

Do you mean Future Perfect? This one is missing the verb "go." Future perfect might say something like:
I will have gone to bed by the time you arrive tonight.
This reflects a situation in which you are "projecting" yourself into the future (specifically, the time that you arrive tonight) and you discuss an action that will be in the past at that time: specifically, when I knock on your door at 10p tonight, you will have gone to bed sometime before that time.

Future tenses don't get tested a ton on this test, though they can be tested. Present perfect and past perfect are probably the most frequently tested.