AAA0914

Yes, they should be separate checks as they are separate filings (i.e. if you misfile the I-485 and it gets rejected, the I-130 can still be accepted).

No, it is not a form. DS3035 transforms into I-612 just because DOS and USCIS work differently and process it differently. Your approved waiver will say I-612 application to waiver foreign residency requirement but that's all. You don't need to do anything else at this point, just wait.

You don't need to file anything else. I-612 is the form assigned to you at the USCIS stage. You just need to wait for them to approve the waiver and send you the approval notice.