1.) Well let's make a deal :) because it is a loose/loose situation for you. Let's make a win/win situation for you :). Find a project(doesn't matter big or small) and once you get a project then ask your employer that current client wants to hire you. They will say "NO" first . Then tell them that you can give them some money like $1000 and let you join the client otherwise tell them that you have to join another project and other project is already in your hand and you have to switch the employer. Well they can't say No to free $1000 :). This is just my 2 cents and it worked for me. You are a GC holder man try to use it on your behalf. Sky is your limit
2.) I don't think client will conduct background check. You can ask them like "What are the hiring process and paper works? Does that include background/criminal/reference check? In my case, they didn't
3.) To be frank, Most of the client knows that all the things which you put in your resume are not true so they don't care and don't go into details. They just want to hire you because of your work. Think in this way, why do they want to play a game with new guy who they have never worked with rather than hiring a person (you) who they already know and saw you actually doing good job.