Great questions!
1) Spouse not on lease: For us, any applicant who tries to negotiate any changes to the lease or rent is no longer in the running as a tenant. This would only be the beginning of this tenant taking advantage of you. Next, they'll paint the house without your permission, and demand that you pay them for the paint. They'll have one of his buddies living in the garage because you don't require that everyone is on the lease. What if the wife damages the property and refuses to pay the damages? You would have no recourse because you have no contract with her. Run from this applicant. Fast.
2) Children on lease: Our lease requires that they list the minors living in the house. You need record of that only so they don't take in their cousin's daughter. Then their niece. Then their neice's friend who's parents are mean to her. Etc- next thing you know, your two bedroom apartment has eight kids living there, writing on your walls and dropping cheerios down the vents. Yes- if the "kid" is 18, he's on the lease. He's a legal adult and living in your property, for which you are liable. Protect yourself at every turn.
3) Income requirement: I'm actually much more flexible on this one. If the income is close, but I get really good vibes from the tenant, they have good credit and good references, I am okay with that. I put a lot of stock in their references, especially from previous landlords. I woudln't ask for tax returns, just make sure you have a very thorough application so that you have all of the info you need if things go south.
Good luck out there!