What’s suddenly changed? Has the dog been behaving differently since you moved in together, or did your husband not realise the hair, smell and mess dogs create? Did you have the dog when he lived at yours or is it new?
Tbh I’d try to compromise over the dog as not everyone likes animals in the house. Can you have set rooms the dog is allowed in and dog-free rooms eg no dog allowed upstairs or on the sofa?
I guess before the dog was in your home, but now it’s half his home (presuming he put an equal amount into the purchase) he wants to set boundaries? If he’s sold his place to buy one together he’s got nowhere to retreat to. And I think you both have a say when it comes to things that bother each other, you’re married and now jointly own the house.
Is the dog house-trained? Noisy? Over-excitable? Find out what his issues with the dog are so you can solve them. Eg if it’s making a mess on floors he’s just cleaned, wash the mud off outside. If it’s smelly get it washed and groomed more often and clean it’s teeth. If it’s behaviour annoys him, get some more training and teach dog what’s acceptable to both of you (eg if you feed dog from the table, stop doing it and feed dog separately).
Re your son, it’s tricky as he isn’t your husband’s son and he’s an adult. What does he mean by ‘guest’? I think it’s fine to change son’s bedroom into a guestroom (my mum changed all our rooms around once we went off to uni). We just had a cupboard with our stuff in. One room became the guestroom, another her sewing room, another a study with a sofa bed.
Has your son changed since being at uni and your husband wants to put boundaries in place? Eg if he’s staying the whole holidays every term maybe that’s a bit long and he needs to spend some more time at uni or get a holiday job? Is he staying up late gaming or leaving a mess that your husband objects to?
Before assuming the worst I’d try to work out what the actual problems are. Living together in a house you both pay for, even if you’re married, is an adjustment and needs compromise.