Visa and funding options for undergraduate internship in US I wrote to a professor (let's say Prof. X) in a US university (cannot disclose the name) for an internship over the next summer. Prof. X is happy to have me work with him. 

The university has a program for interns (that takes students through the J1 visa program) but the dates do not match with my home university's summer break schedule and I cannot overstay. I asked Prof. X about this and whether I could work as a student independent of the program. He's not sure if he can help me with the visa process if I don't come through the program but he said that he's looking into it. I also sent a mail to the program manager's office 2 weeks ago but they haven't replied back. 

Also, I cannot apply for a J1 visa without a source of external funding. My country does not have any government scholarships to fund such research internships. I'm unsure of asking Prof. X for funding because I don't know the proper etiquette in the US and if it's polite to ask for funding. I could make do with personal funds, but it would be a big help if there were some external funding. 

My questions -


First and foremost, is there any way I can help Prof. X with looking into possible options? I feel bad about not being able to do anything from my side right now.
Should I ask Prof. X if funding is available? If it's not impolite to do so, how should I phrase my request? 
Are there any scholarships in the US that I can apply to that would fund me (I have a very good resume)? 
If the university is not able to provide a DS2019 for the J1 visa to students wishing to work independent of the program, what other visa options do I have and would they also need external funding? I know that the F1 visa is not an option as that requires enrollment in a university. 


Edit 1 - My home university's summer break starts and gets over a month earlier than the specified dates of the program at the US university. I will not be allowed by my home university to register for a semester more than a week late.
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.