If the Supreme Court had wanted to get involved in these cases, they would have. It's impossible for them at this point. if forced, Obama would win 9-0.
I believe that the House could impeach a president for being unqualified to serve
Yes, they could. You are absolutely correct. They would, however, have to explain in the proceedings why they let him be president in the first place. This could happen if new information came to light, but there's absolutely nothing new in the case against Obama's eligibility.
If the Supreme Court had wanted to get involved in these cases, they would have.
That's not how courts work. Someone has to bring a case to them and it has to be within their jurisdiction.