Private caregiver for veteran. Do I have to fill out W2 and background checks and all that?

If you're being paid by the VA, I think you better forget about doing something under the table. The VA is going to keep records and probably will either send you a 1099 (G?) or take deductions. In fact, if your checks reflect deductions, then I would think that you're an employee of the VA.

I get that but I'm retired and just do this part time under the table because I raise my TWO grandkids.

You need to raise these issues with the VA.