It's easy to bring dogs & cats to Thailand as long as it is not a banned breed.
As said above to return to Europe your dog will require a rabies titre test that can take up to 8 weeks to be processed along with all mandatory jabs. If vaccinations are kept up to date to the letter some (if not all) EU countries will not require a new titre test with each entry. It's best to do the titre test a month (not earlier) after the rabies vaccination to have higher titre count.
Sorry, can't help with cat but would think it's similar.
It indeed appears that Thailand does accept emergency/ temporary passports as long as they are issued by a country within the visa exempt scheme. See point 3.