Visa and funding options for undergraduate internship in US
I doubt that there is a lot that you can do. A lot of the options will depend on the internal policies of the university and/or the department concerned. Since your professor is better placed to contact the department administration, international students office and any other entities involved in the process, him/her coordinating the process would result in the most efficient resolution of the problem.
Absolutely. You only stand to gain if you do. Professors understand that it could be expensive for international students to fund themselves and will do all they can to help you with it. Its best to phrase your question in a brief, straightforward but polite manner. 
A lot of the scholarships require a US citizenship or permanent residency. Your best option would be to contact your home university, destination university or your home government to look into options to sponsor your trip.
As far as I know the F1 full time enrollment requirement is for the academic terms and does not apply to summer programs. The best judge of this would of course be the International Student Office at your destination university since it is a matter of university policy.