First day as an independent study student
I think you're wasting too much time psychoanalyzing your professor. If the semester has already started and he has not reached out, it's fine to e-mail your professor to set up a meeting.

I will advise hold on to the problem set answers for now. In the meeting, you can definitely mention the prep work you have done and you're enthusiastic about this opportunity. Focus on setting up a study plan, including the preferred frequency and forms of meeting, deliverables, and evaluation criteria, etc.