1: visa exempt and visa on arrival are 2 different things. Visa exemptions can be extended for 30 days.
2: No, most offices let you do an extension 30 days before you current stay expires. Bangkok, CM, and Phuket are 45 days. Pattaya is like 5 days. The 30 days is added to the end of your current stay, not from the day you apply.
It's not just 'against their beliefs', it's against the law in Thailand to kill a dog. The only time a vet will/can euthanasia an animal is if they won't recover.
Glad you've had positive experiences. Some DLT offices (like chiang mai) require forign sellers to be present at the DLT when transferring as well as a certificate of residency (yes, from the seller). I'm not saying you can't, it's just more of a pain than if it's registered to a Thai.