Many people treat their pets like children, and while joint custody of a dog may seem odd, it’s actually a situation that some divorced people find themselves in, so that’s not necessarily a yellow warning light.
It is odd that no papers have been filed in your boyfriend’s divorce. In fact, many people say they’re separated because they are living separate lives from their spouses, but the legal reality is that depending on what state you live in, legal separation papers need to be filed with the court only after which time (sometimes six months), a divorce may then begin to be granted. So educate yourself on divorce in your state, and the state of your boyfriend’s divorce before you invest any more of your own heart in someone who is still legally married.