You can't say "I went study". You could say "I went to study...". You can say go and study, often in the sentence: "I'm going to go and study..." It is very common to say: "I'm going to go and.... e.g. ... Wash the dishes. ... Clean the bathroom. ... Have a shave. ...play Fortnite. etc...

No. A phrase like “go study” is just short for “go and study”, so both verbs are in the imperative mood. It is *not* a case of a conjugated verb governing the base form. There is no past imperative mood in English, so the construction cannot possibly be used in the past. It is possible however, to use “go study” as a compound infinitive (preceded by any construction that governs the infinitive or bare infinitive), for example: “We wanted to go study Chinese at the Chinese school.” Note that some pedants might claim that "go study" is ungrammatical and requires the word "and". Don't believe them, as the form has been around for centuries (and even predates the English language).

Correction- “I studied this morning.” Or you could say “I went to the library to study” You cannot say “I went to study.” People will still understand what you mean, but it’s not proper grammar.

It is more "I studied this morning". Or, perhaps, "This morning, I went to the library to study".