You can work remotely anytime even on visa exemption or a regular tourist visa. I mean, if you're in your hotel room or condo and doing some work for your employer on your computer: what's wrong with that? Who's gonna stop you? Heck you can even take your laptop to a cafe, sit there and do something on your computer. Again: what's wrong with that?
I've never heard ANY visa agent say anything positive about the DTV. Anybody, from any country, can apply for it, anytime, while NOT in Thailand without the need for an agent. That's why they hate it! People who use agents for DTV are just lazy or stupid.
I did receive an email containing the payment receipt. On the application site you can also check the status there: pending this, pending that...you may or may not get emails before the final result.
Your biometrics (face photo + fingerprints) will all match one person, regardless which passport that person carries. So, immigration will know you hold more than one passport. As to what Thai immigration says about it: I don't know. Good luck.