It’s hard to tell from the pic cause I can’t get a good view of the leaves, but based on the bark, it looks like maybe it’s in the willow family. I’m thinking populus deltoides (cottonwood), but I could be way off. They often have a lot of dead branches, especially in the rocky mountains. It seems like your neighbors tree might have broken a little bit in that crotch, allowing infection to get in. It seems like there is a bit of slime flux causing that staining. If it’s close to housing, it is worth having an arborist look at it.

Like someone said below, do some due diligence to make sure you are getting a certified arborist cause otherwise, you’ll probably just find a “tree guy” who will automatically just say it needs to be trimmed or taken down. There’s a decent chance the two sides of the tree could be bolted together to reduce stress on that joint.

In don’t think anyone can tell you from a photo. You need an arborist to look at it.

You can talk to the neighbor and express your concern. Ideally, have a paper trail in case something does happen.

You can have an arborist trim any branches hanging over your property without asking the neighbor.

I would get an estimate from a couple arborists for the trimming what’s over your yard. Then, approach the neighbor and offer to pay that amount towards having the whole thing trimmed—must include trimming the parts over your yard.

I wouldn’t pay until the work is done.

If you don’t know the neighbor at all and/or can’t be sure they’re trustworthy, I’d just get the part over your yard done.

Split trunks like that form a weak spot for rot, but you’d be better to have an arborist look at it.

It can be corrected by drilling a bolt through each trunk and wiring them together so each is supporting the other instead of leaning and pulling apart.

https://www.heartwoodtreecompany.com/blog/how-to-fix-splitting-tree

It will live a lot longer then all of us and many of our descendants

It’s fine. Worst case you pay a deductible and get a new roof.

Maybe?