This is a tricky one. But some of these just don't sound quite right.
Over this year, he's been able to provide a lot of correct answers in class.
I would change "over this year" to "over the course of this year" and I would use it to describe a process, such as:
Over the course of this year he improved in his ability to provide
correct answers in class.
That suggests that the improvement was gradual but observable.
During this year, he's been able to provide a lot of correct answers in class.
This sounds right to me. It happened during the year, or throughout the year, on a day by day basis as the year was lived. It's just the way we talk where I am.
In this year, he's been able to provide a lot of correct answers in class.
This doesn't sound right; it's the kind of thing that gives you away as a non-native English speaker. "In this year" or "within this past year" is usually used to indicate when a certain event happened.
It is not used to indicate something that happened on a day-to-day basis. For example, one person might ask another, "When did they get married?" The answer might be, "They got married in this year, during the pandemic when only a few people and the minister would attend the wedding." I think that doesn't sound quite normal either and that people would say it that way only if they wished to emphasize how unhappy they were with the situation of the pandemic and lockdown. Otherwise, they would just say "this year."
I'm not sure if this answers your question. Nor can I think what more to add so I'll leave it at that for now. Let me know if you have more questions.