Do I need my employer's company to be registered in Laos to apply for the DTV as a remote worker?

Not only is there no issue, it's actually better to work for a business NOT registered in the country you're applying.

I've heard of some consulates asking for things like tax returns or official proof of being on the payroll (in the case of Burmese applicants applying in Myanmar on the basis of locally based companies).

This won't happen if you're working for a company in a third country.

My business involves remote work in Myanmar and Laos, although the bulk of it is in Myanmar. Partly for this reason, I'm not applying in Yangon as I don't want there to be questions and additional documentation requested that I may not be able to supply..

Myanmar being in civil strife and with the complications arising from land registration in marginal areas (and I'm not even talking about anything involving foreign nationals here) it could get quite complex.

Of course I may be overthinking things here, as I only deal with Burmese stakeholders and clients, with the company I'm working for being registered outside of the country.

Still, it's better I apply in Taipei or Indonesia as they're more flexible.

No issues. Just make sure you have a letter from your company stating you can work remotely or a contract that states this. Highlight it and submit with your packet.

You can apply in many different countries in that situation.

It doesn't matter where the company is registered or located (except a Thai company of course or registeredin Thailand) as long as you can show that you're allowed to work remotely and be able to stay in Thailand without any issues.

My company is in USA and registered in USA, I flew to Taipei from Thailand just to apply for the DTV, then flew back to Thailand the same day that I received the DTV.

No need to

Does not matter, but you will need some kind of service contract.