I know I jumped on you in another thread. I don't have info about getting a SSN, since the last time I applied for one was in 2001 while I was on a temporary visa (H-4), and that application was rejected, but I do have info on going through criminal background checks.

What criminal background check do you have to go through? If it's just your typical police clearance from your city's PD, you don't need a SSN. You can use your state ID (if you have one) or your passport. You don't need a SSN either even if you're getting an FBI clearance done.