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 -

 1. 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.
 2. Should I ask Prof. X if funding is available? If it's not impolite to do so, how should I phrase my request? 
 3. Are there any scholarships in the US that I can apply to that would fund me (I have a very good resume)? 
 4. 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.