My situation

Legalmatch gave you some good information but I want to add to it.

North Carolina recognizes legal separation, that said, in the separation agreement spousal support and child support can be set up - and an agreement made on other bills if necessary.

I would rather you don't even consider trying to take custody of the child until you know for sure your marriage is going to stay intact. As LM said, it would be difficult for you two to be awarded the child in the first place - and it wouldn't be fair to put the child into the middle of a rocky relationship when the child has a stable life with mom right now.

I strongly STRONGLY suggest marital counseling.

On other questions, no there is no time limit as to when you can get a paternity test, unless there is a court order for it -  and as far as payment, it will more than likely be split between the two of you.

One thing you have to come to terms with - the whole "she just wants our money" thing - TBF, the children your husband fathers, be it with you or anyone else, deserve to be supported by both parents. She isn't doing anything wrong by requesting support. She as a single mom probably relies on his help - think about it - you are worried if you separate how you would pay the bills - she's actually IN that boat. The only one who suffers in the long run is the child if they aren't being supported.

For your husband the get visitation, he has to file for it. She has the legal right to keep the child from him until he takes action to change that.

Don't get into it with her over the phone or e-mail or anything else. It can complicate his potential custody if you are nasty enough in your communication to warrant a restraining or no contact order. Direct your anger where it's due - at your husband for stepping out of the marraige and creating this mess. She's just a mom fighting for what is right for the child she is caring for 100%.

I am sorry for your situation, I can only imagine how hard it would be to hear that news and to now have to deal with the legal issues surrounding child support and child custody.
My immediate advice to you is to stop sending email to the other woman as she will likely bring it up in court and you will settle NOTHING emailing her, whether they are nice or mean.  You need to remember that this is a mother wanting money to support her child.  If your husband is the child's father then she is entitled to child support and the fact that the child was conceived through infidelity has no relationship to his liability for payments or the amount he will pay.  Unmarried fathers child support duties are just as enforced as once married fathers.  If you have concerns whether the child is his then a court-ordered paternity test will answer the question, and is probably the first step in the proceedings.  If it is not his then the fact that he as been paying child support has no bearing on continued payment and it would be up to the judge whether he can get his money back, although that may be complicated if she does not have any money to give back!
Your questions regarding whether you and your husband can seek custody are a judicial decision but yes the option is there, however courts are reluctant to award out of state custody to a father that has had no contact with the child for 2 years. 
Situations involving child custody or child support are very court intensive.  It may be wise to talk to a family lawyer to discuss the best strategy to protect yourself and your children.  Consulting with an experienced family lawyer can help you understand your rights and help you deal with the complicated court system. Good luck with everything.

NC is one of the 6 states where you can file for alienation of affection. You can sue her for every penny she has, plus penalties. These almost always succeed.

I met a guy recently who said that 6 years into his marriage he and his wife agreed that they didn't have anything in common and the marriage wouldn't work. They had a 2 year old so stayed together for the sake of the child. They were civil but didn't love each other. She had an affair, he had an affair, they split when the girl was 13. That, he said, was far worse. If the girl had grown up without both parents around it would have been normal. He said he wishes they had split earlier when it was easier for the child. At 2 you don't know much.

Just a thought.  A lot of people on here were cheated on, forgave, stayed together, only to have it happen again years later.

It may be that it will work. It might not. If the only reason to stay together is for the child, I don't know.

I wish I had left my husband years ago.

well I am no lawyer so I would ask the spazater. Just me being a man It makes me wonder about your Husband but hey chit happens. So the only concern I would  have would be with me. Also in your state or the state she lives in can you sue her for alienation of affection some states allow it. That maybe an answer to you wonder what your going to do. I would ask the spazster.

Also, we already have 2 girls together and she and I were pregnant at the same time, me first. Is there away to separate without getting a divorce? How would I pay the bills if he is the breadwinner? Would this just "mess up" the kids and bring the problem to the forefront? Right now, they have no idea because we have been being very civil and talking when they are not around. They are 2 & 5. I feel like I can love him again, but I also need time to heal. He is doing everything "right" in the aftermath according to everything I've read but there's been so many lies before, how can I bring myself to believe that the same thing won't happen again later on down the road?