I personally think it would be a mistake to marry someone you are so uncertain about. I realize you were not saved in the beginning, but if you had married him then, when he was being so wonderful, would you have felt it necessary to stay with him, when he got negative and miserable?
You have a dilemma, there is no doubt about it. But only you can make this important life decision. One thing I can assure you of, it that each person has to follow Christ, what happened with their parents or grandparents does not affect their eternal destiny. God calls us as individuals, not as the children of Christians. Your children would not be "illegitimate" if you married, although these days, that doesn't seem to be an issue anymore, does it? As far as salvation, that is in the hands of God, and their legitimacy status does not affect it.
The fact is, the world lead you into this problem, with its lack of respect for marriage. The world will not lead you out, but it will lead you further astray. By consulting with Christians, you are making a good step forward. However, I see some good advice and some bad advice here. You do not HAVE to marry this man. You need to figure out what is best for you and the children. You need to examine what your commitment to this man was in the first place, and how has it changed?
Finances should not be the sole determining factor in deciding to marry him. That will only breed resentment and bitterness if you do marry for that reason. Because if he becomes more depressed and angry, and becomes abusive, you are really going to regret that you got trapped into a marriage that turns out to be more destructive than good.
Being sick adds another dimension to this issue, especially so young, and with children. What is the prognosis for your recovery? Do they think you will be able to get the Lyme out of your system? Or will you have problems the rest of your life? I have a cousin with Lyme disease, and she is very disabled. Her husband basically has to do everything. But she was not initially diagnosed correctly, or treated properly, your situation may be quite different.
In the end, this is more complex than some opinions here, or a list of pros and cons of your boyfriend. It really troubles me that he is angry and that he doesn't respect you. (Thinks he needs to make all the decisions, did I read?) Because if he is this young, it could get worse and worse.
With respect to the illness, many people don't handle their spouse or SO being sick very well. My husband actually become quite depressed in the early years I got RA. And I was depressed, so it was not a happy marriage, at that point. But, having been married for 20 years, with 4 children, divorce wasn't in our future. And, we did walk through it and out the other side.
The point being, you have no marriage commitment to keep on pushing through the hard times. You have no contract to stay together, in spite of your sickness, or your lack of contributing to the finances. I know that it something you cannot change. But, perhaps you need to work through what you want from your boyfriend, before you are willing to marry him. And you should marry him if you continue to sleep together and live together. And I know it is hard to walk away, given your circumstances.
Sorry I could not be more help - just trying to give you some feelings and thoughts about this difficult situation.