Do you let your LO return to their house to go through their belongings?

I think this is a very individual and instinct kind of thing. Some LO's are going to be benefited others are not at all and I can't make that judgment for you and your father since I don't know and love him. You and maybe your siblings depending on circumstance know him best and when it comes to it need to try and remove your emotions from the equation (it may be sadder for you than for him right now that he missed Christmas dinner), tap into your love for him and decide what will benefit him most. That said might it help at all if you just said to him, Oh Joe (generic you insert the name) finished that project or you and Joe took care of that before you went to the hospital remember? Maybe if some of the things he wants to go through could be brought to him one load at a time, say a box of old photos for him to sort, at the AL and when he finishes that another load could be taken over. You could simply say the reason for this is that there is a lot of coordination to get through the process of taking him back to the house for a few hours or something so you brought stuff for him to start on while you work on that. Is he going to be living with Mom again now that they are both in AL or are they fairly independent of one another? I ask because that could be another reason for him to stay at the AL facility, to help Mom and keep her feeling secure because going back to the house would be too hard on her. This way it isn't about "him" it's about what you and he can or need to do to care for her.

I would not let him go home. As already mentioned, he won't want to go back to AL and this will be tough on him and you. Why put him through that? The "projects" are just excuses and with dementia he likely won't remember anything he'd previously planned.

It varies. My in-laws are going through that transition right now, and we're clearing out the house.

My MIL is physically and cognitively able to go there whenever she likes. She has no dementia. She does mourn the change, but accepts it. Still, we do have to watch to make sure she doesn't overdo, so the heavy work is done while neither parent-in-law is there.

MY FIL definitely has dementia. He LOVES to overstretch his limits, and to attempt to smuggle items like, oh, chainsaws(!) back to Independent Living. (Every time we think we got all of the dangerous stuff off the premises, there's one more item lurking in a corner.)

Hence, we are now taking MIL for light visits only, and keeping FIL out of the picture.

My husband is also slightly disabled. So, we who are not disabled, and can still plan and can drive, try to work our schedules so we can work unhindered while granting the less-able their dignity.

Sad times. I'd never pictured this, but that's where we are.

I would not bring the LO to go back to the house. Sounds like it would be too difficult for everyone involved.

It was a battle to get my inlaws into a care facility and we knew if we brought them to the house to "go through things" they would not want to back or it would be too emotional for them or they wouldn't want to give up any of the items. We had a giant sale of their household goods because they didn't need the stuff anymore but did need the money. I took pictures of the stuff in their house so they could point out what they wanted to keep (which was a lot of stuff). My MIL was adamant about coming back but I just kept refusing. I don't think it's a good idea to bring your dad back to the house. Projects? No. There's no point in keeping this delusion alive. Also, tell them a "friend" of yours is taking care of the cat.

I know this is all very difficult. Part of dementia seems to be wanting to go home. Sometimes the person is already at home. Sometimes they are not referring to their most recent home but their childhood home. It seems to be a desire to be somewhere besides where they are. Most people advise not to take them home but to divert their attention to other topics. Some think it makes it more difficult for them to adjust if they go back and forth.
There is a sometimes poster on here who would take his dad home and even on fishing trips when he first went to an ALF. He has since moved to memory care and had suffered a fall and broken hip. His father has vascular dementia. His name is Linkabit if you want to do a search for his thread. He made it work for him for a good while but I think that is the exception to the rule.
I would want to take him home if it were me but I think each person has to evaluate their own circumstances.
The death of their cat makes it more difficult to take him home as well. Not sure how you would get around that. It’s a tough situation.