For example, as a student, I have a "general" or "generic" question. My question is "Does every student need to wear school uniform every school day?" Are "general" and "generic" interchangeable here? If so, which one is more common in colloquial AmE?
In my experience generic and general are not interchangeable here or even in most cases.
When you say "I have a general question," you are saying your question applies to a broad topic.
If you say "I have a generic question" it means that your question is a very plain, common question. It is a question that you would see asked every day. It also carries a slightly negative connotation because it's like saying "I have a question that isn't unique or creative at all. It is average"
So general is the correct usage here.