Unless you are leaving out some kind of medical issue, it should be safe. If you are worried it may be too soon, only you can know that. As a dog owner, and a friend of many other one, sometimes the best help in healing a broken heart from the loss of a pet is a new pet. It won't replace the first one and it won't fill the hole in your heart. It just make the heart grow bigger so the hole doesn't seem so big.
It depends. If the 13 year old is not interested in maintaining a Shih Tzu's coat, don't get one. If The 13 year old does not want another dog, you will only make the grief far worse. Casually bring the topic up in conversation or something. See what is said.