1. You need receipt number.
2. Don't know. Ask your lawyer.
3. Possible not. I met once. I believed international service of our University should send 129 and 539 together using premium process. However, I cannot sure since those guys are very stupid. But my wife's 539 approved is 6 month later than my H1 approved.
4. I think you need let your lawyer or international office do it. I don't think USCIS accept personal check.