This thread will probably be moved to another forum, but I can answer some of it.

You are correct. Eurostar may not check the name on the ticket. However, the French border control (which you will encounter at St Pancras) probably will. I remember handing over both my boarding pass and passport to the agent (who, after a few seconds to check the docs, stamped my PP without a question).

I cannot answer about 3rd party Eurostar tickets, but keep checking the Eurostar page. I found prices tended to fluctuate on a daily basis. But since you are within 12 days of departure, there's probably a low chance they would go down. Good luck.