It is true that only tourist visa holders will still be able to get covid extensions. Or rather non-immigrant visa holders no longer are. It would be my understanding though that an STV is in fact a (special) tourist visa. And hat is how I would argue my case. When in doubt: try a different office / officer. The inconsistent outcomes always surprise me.
Vietnam is open with just a test (PCR 72 hours / ATK 24 hours). Even visa free entry is reinstated for certain countries. Returning to Thailand would be the bigger pain in that case.
You came with an STV? Why did you opt for covid extension then instead of regularly extending your STV for another 3 months? Anyways: Did they give a reason as to why another covid extension could not be done? I had a covid extension ending on March 25 myself and got another one until May 24 now. Initially they said cannot. Then they wanted to give only 30 days. Too many immigration officers just don’t know the rules properly. We explained it to them and walked out with what I wanted - and what the current legislation allows: another covid extension.
Everything in Thailand can be done two ways: Officially / yourself or through some (sometimes shady) agent. I was referring to doing it yourself. This would involve going to the Department of Land Transport. For conversion, they might require you to have a translation / confirmation of your national drivers license by your embassy (costly and annoying ~1800 baht) + residence certificate (500 baht) + watching 1 hour video + paying around 300 baht for the actual Thai drivers license. Due to inefficient bureaucracy, be prepared to lose at least one day over this.
Unlikely. Only the issuing country of your national driving license can issue those. I doubt they would issue you one remotely and send it to Thailand.
You could alternatively consider getting a Thai driver license. They are rather easy and cheap to come by. Either make a complete new one, or have your national license converted. The latter might involve quite some bureaucracy.