I'm sorry that Puddy is still hiding so much in her carrier. It must be her safe zone and feels better for some reason. Maybe she is hypersensitive right now with all that is going on and noises and light might make her feel worse? It's hard to know. If you have a few extra carriers around, you could put one in a few different rooms with soft bedding so she has a variety of 'caves' to choose from. That way she could at least be nearer to you. It sounds like the one in the basement is her favourite though.
Prayers going out to you and Puddy and to Boo! I hope that his thyroid condition will be manageable so he can get healthy. You sure have a full plate right now!