Agree with the posts above. It's a straightforward list of requirements for a tourist visa, but requires attention to detail and the wait for an appointment is a hassle. Still very doable though, and know a few people who have sailed through the process in the last few months and managed to get 10 year multi entry visas out of it.
Weighing in on the pet aspect: I brought my dog in on a visa exempt stamp earlier this year and didn't get a single question about my status. As long as the pet paperwork is in order, I think your own visa status isn't relevant. I would definitely recommend getting an experienced pet agent to help make sure you have the right paperwork and carrier, etc, for departure from country of origin/ getting the pet on a plane, but the arrival process was painless, just filling some forms and paying an inspection and customs fee.
Your dog can probably fly with you as excess baggage. That should cost a few hundred dollars, plus the cost of an airline approved crate, plus getting him export ready/vaccinated/other paperwork will add a few hundred more. Entry into Thailand is a few forms and a few thousand baht in fees and customs duty, easy. It can be worth paying an agent to support you in navigating the export process but you shouldn't need to send your dog by cargo, which is very expensive. If you can manage on your own it would probably run about 1k max, including all the vet visits etc.