I'm still so sad for both of you. Poor little hero, I hope he feels better soon.
As for your question on dogs grieving, I don't know. Like humans there is no specific way animal will mourn.
I'm not saying rush out and get another friend for Tully, especially if you aren't ready yourselves. But when Shadow's brother Buck passed he only cheered up some when I brought home a pair of puppies. I do not know how long it would have taken to work through his grief if I hadn't gotten him new friends but it had been about two months where he'd nearly stopped eating and lost 20 pounds.
He was never as close to them as he was to his brother and Bear and Pete weren't all that close to each other either. They didn't mourn when Pete passed, Shadow was too addled with dementia when Bear passed, and now they and Buck are together again.