Quote from @Dennis Adams:
Thanks. I'mprobably overthinking it - just would really, really hate to evict someone for any reason and haven't had to so far. I'll go ahead and verify everything and if it checks out, I'll rent in his name only. Never rented to anyone without an income that wasn't a child and really hadn't considered it - strange as that sounds. I appreciate everyone's input.
Hey Dennis,
I mean this seriously and with kindness - you may want to hire a property manager. If you cannot evict a family, you should not be a landlord. I'm not saying that you look forward to it or hope for it, but you've got to be capable of doing it. Otherwise, someone decided that you'd be a charity for them. You may be in a financially different situation than me, but I'm not in a place where I can pay for housing for strangers indefinitely.
You will want everyone on the lease. If you leave her off the lease, you could potentially have a squatter if the person on the lease leaves.
As everyone else has said, their kids and marital status aren't your concern. Their credit scores, income, and references are. The two addresses are important if you're checking landlord references, but not as a way to dig into their status.
Having a property manager will take the liability off of you and prevent you from having to do any of the "unsavory" parts of managing property. There is no shame in having someone else manage the property for you. Having rentals is an amazing accomplishment and that is not undermined by having someone else manage. When a person owns a house, that isn't any less awesome if they hire someone to cut the grass or clean the bathrooms or paint the walls. :)