I brought 7 months supply to Thailand and some of it was gel. No questions were asked. All were sealed with prescription info attached and in sealable plastic bags. 6 of them!
My experience is that you can buy some medications much more cheaply from a hospital after seeing a doctor than you can from a pharmacy. One of mine was less than a third of the price.
As for the dog I would leave it at home as in my experience the local dogs here would not tolerate it if they came face to face at any time. Many dogs here are not pets and will fight to protect their territory.
I think there has been a misunderstanding. I am talking about 14 days quarantine in an approved hotel after entering Thailand. Where does the hospital come in?