It sounds to me as if you have a good handle on this, you are just feeling insecure about it. He hasn't been around for 2 years, he hasn't paid child support, and suddenly he's decided he's a dad and now he wants to pay his attorney to ensure he gets to visit his kids ?
Every state is different. I got divorced in Maine, and in Maine, because my ex and I got along, our divorce decree simply said, "reasonabls visitation", and we had to iron what that meant out ourselves. It worked until he remarried, and then he ignored the kids, so when he did show up there were LOTS of emotional issues going on which made our lives stormy before and after the visits. (The emotions always come out in the relationships where the children feel safe, so you are going to have to do lots of mopping up. They will probably be on their best behavior with him, 'so Daddy will love them', and the hurt, anger and insecurity will come out when they get home. It's not easy to deal with, but the good news in there is that when they act like that, you can remind themselves that YOUR residence is where they feel "at home".)
One day a week is okay, but you may find it imposes upon the time YOU want to spend with your kids. Most visits are every other weekend, I would think that going from NO visits to weekly visits interrupts your family often enough that things are going to be very very unsettled at your house. When my kids did go to their dad's, (which wasn't often) it took a whole week for the kids to settle back down into life in their own home.
I would say that the legal document should provide for no MORE than once every 2 weeks, and leave room for him to back out again or to allow you to agree to longer or more frequent visits at your discretion. If he left them for 2 years, the reality is that he will leave them hanging again. While he is interested in having a relationship, you can't really deny him one unless you can prove that he is abusive, neglectful or otherwise a danger to them, but I would be inclined to fight for my kids and say one Saturday every other weekend. Especially since they can't stay at his home anyway. In the end, they will most likely be going to visit Grandma and Grandpa, more than their father who may or may not show up.
What I learned about divorce, in retrospect, is that I should have been less "kind" and believed in myself more. I wanted things to go smoothly and to ensure that I backed down more than I should have. It sounds to me as if you are intimidated by him, and more particularly by his attorney. But you are the one who has been responsible for and faithful to the children for the past 2 years. If it had been up to him, they'd be in foster care. Walk into that meeting remembering that the ex and the attorney are only men, one who wants something from you (because he willingly gave it up in the first place) and the other is merely doing his job. Nothing more. Don't be intimidated by the courtroom setting or the formalities. Do what is best for your children, because you will be dealing with the fall out of this for the next 13-20 years.
Down the road a piece, you might find you gave him more visitation that you should have. Or you might find that you denied him visitation opportunities you wish you had allowed. Which situation would make you feel worse? I figure you can set a maximum court ordered amount, but allow wiggle room so you can be more lenient on a case basis, depending on how things are going.
The other option is to make visitation a very specified option now, and when he wants to increase it, he can return to the courtroom to discuss it again. That would be the more expensive route, so it really depends on whether or not the two of you can get along and deal with the issue together, or whether you need to have the court standing guard between you. Honestly, the more you two can work out without lawyers and judges involved, the better off you will be and the children will be, because you guys will handle it all better. But if that won't happen, by all means, nail it down tightly -- but don't be surprised if he disappears for another 2 years or so at some point. He's already shown a history of walking away.
Good luck! When you drive there, imagine it going well, and imagine yourself standing up to them the way you, as a nurse, would fight with a doctor to advocate on behalf of a patient. (!) You are strong and capable and you want to be thinking of those kinds of things rather than walking into the debates feeling worried and insecure. Be honest, be caring, and reserve for yourself the family time you want and need to provide a secure home base for your children. When you walk into the room, breathe deeply and slowly, be calm and do your best, with whatever resources you have within you at the time. That's all any of us can ever expect of ourselves and each other.
Kudos to you, Mom! It's a hard job, and you've been handling it and your nursing career for 2 years now. You can do it ! :-)