H.,
My heart goes out to you! I was in your daughter's shoes once, when I was 6, and I was in your shoes just last year. First let me say, you are a fighter! You had cancer and overcame that, and I'm sure your kids were your inspiration. I know you don't want Emily to resent you and your worried about her dad taking her from you and all sorts of things are running through your head at 10,000 miles an hour. You mentioned that you didn't know whether or not you should go to court or try then fail. You're not going to fail with whatever you decide. The reason: your kids. You're going to make the best decision because of your love for your kids, plus that, you're a survivor! I say, let go of the fear and gear up for the war. From my experience, your daughter's dad isn't going to do anything. I bet you this is his logic: If I threaten H. enough about taking Emily, H. will let me do whatever I want and I may even get her to take off the child support, which is most likely his sole interest. He hasn't seen Emily in 2 years, now out of the blue shows up and wants to play daddy. I don't think the judge will buy that story either. The only way you can get complete custody is if the father is of harm to your daughter, and you can prove it, or if the judge decides to take away his rights because he hasn't seen Emily in over 2 years. Seek legal advice, make your decision, and go for it! If Emily's dad thinks that by quitting jobs is going to get him out of paying child support, he is sadly mistaken.
I'm 30 now and my dad pays my mom child support for me and my 2 brothers who are 28 and 31. He owes over $100,000. He can hide, but he can only run so far before it catches up to him. I know how rough it is when your hurting for money and have to do whatever it takes to make sure your kids have what they need. At the end, you'll be on top. Emily is still little, and she is smarter than you may think. My daughter is 3 and knows things I would've never imagined that she would understand. Just be cautious about how you talk about Emily's dad infront of her. She will begin forming her own opinion about her dad in time. All you can do is make sure your kids have everything they need, which is you!
Regarding your health, I think that you can devise a letter yourself stating who you want your kids to live with if something should ever happen to you and get it notarized. I'm not sure that's enough, but it's a start.
I pray that you remain cancerfree for the rest of your life! You are an inspiration to me, and many others I'm sure!
Best of Luck & Love to you and your little ones!