OP you've gotten some good advice. Might I add that as you decide what to do, please consider his relationship with your sons. Yes he is being irresponsible, and should be called on that but it appears he is making some effort to be in his sons' lives. However you manage this, please don't let things degenerate to the point where his relationship with the boys is marred.
Yes they slept on the floor in the house of some lady who is a stranger to you, not what a mother wants to hear, but as you judge him, ask yourself: is he good to your kids? Do they feel like he's being their dad? No matter how bad he is or how we paint him, to your kids, he's dad and they look up to him in some way. The way I see it, kids still love their parents no matter how 'bad' they are, and would rather sleep on the floor while hanging out with dad, than sleep in a lush bedroom and not be able to see dad like they are used to.
As you focus on setting him straight with child support and coughing up the contact information on this woman try not to let your anger damage what the boys have with their father.
You've been through so much already, and I commend you for being strong and a provider for your boys!