What is the name of your state (only U.S. law)? Kentucky

I know I am writing alot but I want to be sure to give the full picture.

I am currently in the process of getting divorced from my husband. We got pregnant by accident, I was on birth control at the time. My husband was very upset at the time of the pregnancy and said very hurtful comments to both me and about our child. He tried to get me to have an abortion and when I refused, he researched ways to force me to have a miscarriage. I kept thinking he was just scared and being selfish. I tried to tell myself that once she was born, he would accept and want her. He told me he was going to a counselor and taking medication because he was depressed. But when I was 6 months along, and I found him in bed with another woman. I walked out the door, and I haven't been back.

Since our daughter has been born, he has been good to her. He spends two days a week with her, and I believe she is well taken care of. I do believe that there is a chance that most of the time he is with his mom so that she can take care of her. But it is my belief that as long as she is being taken care of that it is okay.

When we met with a lawyer, right after our daughter was born, we agreed to pay for one lawyer and do an uncontested divorce. We really didn't have much to disagree about anymore. Anyways, our divorce stated that I was the custodial parent. He would pay less than the state regulated child support, but he would still pay some support, and that we would share custody.

After we met with the lawyer I never received the papers. I learned that was because my soon-to-be exhusband had not paid for his portion of the lawyer fee. Shortly afterwards I learned that my soon-to-be ex had been fired because of his own actions on the job.

The strange thing is that this week he paid the lawyer fees and the papers were sent to me for final signature. The papers were exactly as we had discussed several months ago.

So here are my questions,:
1. Once I sign, does that mean he is legally required to start paying from that moment on? If he doesn't pay what happens?

2. Am I allowed to be concerned about him being a role model for our daughter, and how do I combat it? Is there anything legal that I can do?
* He is currently jobless
* In 2 months, once his lease is up, he will be moving in with his mother, her boyfriend (whom my ex believes has a mental illness. Last year, before I got pregnant, his mother had a restraining order against the boyfriend, and when he broke through the order, she had him committed to a mental hospital. Since then somehow, she has forgiven him and he is back living in the house.), and his mom's two children that are in middle/highschool.
* He is very close right now to being declared bankrupt

I do understand that he is her father as well, and that we are in this together forever. But the man I married was a family man. Believed in fidelity and working and supporting his wife. We were living together for 2 years prior to getting married. I truly thought I knew him. But the moment I got pregnant he began to do a complete 180. I believe now, that he showed his true colors because he got scared that he really was trapped. Every decision he has made since she has been born has been worse and worse. I just don't know how to handle that for her as her father.

From my discussions with my own lawyer, it is my understanding that the law is very clear about joint custody in KY. We are a no-fault state. Unless he would physically harm our daughter, he will be granted joint custody. But what about the emotional impact? Do I have no legal right to protect her against that?What is the name of your state (only U.S. law)?