OP - when I broke my ankle, I had 3 small children, youngest age 3. It was the WORST thing in terms of not being able to do stuff, I couldn't even make myself a cup of tea and walk to the sitting room with it when I was home alone. I have never felt so useless.
Dh had to do everything, on top of working, and amazing friends rallied round and cooked etc to help us out.
Added to that, for the first 4 ish weeks I was in pain. I remember sitting on the sofa, surrounded by stuff to keep me entertained, and just sleeping as my body obviously needed more sleep to heal.
Please be kind to yourself, show you DP this post about having a broken ankle if it would help.
As to the dog, we have a rescue, and they say it takes a year for a dog to settle. We still saw changes in our dog for months after he arrived as he settled more and more, but certainly he will need a few weeks. You are very much in the early days, dog has forgotten dp, and is disorientated and confused, and maybe also in pain.
He will settle, and gradually the 2 dogs will get to know each other. If you go for the stairgate, they can see/smell each other from their own safe zones, and that will help
But, and I can't say this strongly enough, unless his medical condtition means he shoudl be resting, he needs walking. 30 minutes before DP leaves for work, dog walker midday for a good run, and 30 minutes in the evening. Then he will clam right down and probably sleep. If you are worried/DP worried that you can't let hom off the lead, get a retracible one, or a long rope/lead (eg 10m) so he can really exercise safely.