Also, you need to look at the requirements for bringing the dog back into the US, as well. You will want to complete some of the forms before leaving the US
Thailand is on the high-risk for rabies list. Also, CDC policies change often. I believe they were until recently outwardly barring dogs comming into the US from high risk countries.
Once you have traveled to Europe and returned back to Thailand, it becomes much easier to get additional visas, because you have shown your credibility, that you will in fact leave when you are supposed to.
but if you have a digital job & that is how you are funding this, then look at the DTV visa option. It might be less stress (not sure if less cost when you budget the amount of time and other options).
- there is visa exempt (not tourist visa), depending on your citizenship. You arrive and are giving 60 days. You can then extend 30 days for a fee. Then you need to leave & and you can enter and repeat