Learn English – “I am gonna have to” vs. “I have to”

The clause when I first got here is an expression of finished time (cf. yesterday, last week), and requires the past simple. So your first sentence in ungrammatical. If, on the other hand, you use an expression of unfinished time, you usually use the present perfect:

Since I first got here, I have bought a new car, found a job, got married, etc.

The difference is between the words when (finished time - past simple) and since (unfinished time - present perfect).

Could and might are often, but not always, interchangeable. He might have come / studied = 'We don’t know whether has come / studied or not', but you could substitute could. If you want to indicate ability or permission, however, you need He could have come / studied = ‘He had the ability to come / study.'

It's possible that he came from a different culture would be used in considering a historical figure who didn’t seem to fit into his environment. If you were talking about a current situation, you’d say It's possible that he comes from a different culture. It’s hard to think of any circumstances in which It's possible that he come from a different culture would be used, even if you regard ‘come’ as subjunctive.

English modal verbs are both important and subtle. You're unlikely to gain a full understanding of their use in exchanges such as this and if, as I assume, Noah, you are a non-native speaker of English you really need the help of a qualified English teacher.