1- Greg is correct in saying that you cannot be a single parent and join the Army. You wouldn't even be able to get in with just a "family care plan". You would have to actually give up custody of the children to someone else. Part of the paper work you sign while enlisting states that you will not actively try to regain the custody of the children.
However, once you are in and you are at your duty station, you can get custody of them once again, but you will need to be able to provide a short AND long term family care plan. This means that having the children there will not keep you from making it to work on time or keep you from going out to the field, and especially deploying. You will not be punished for fraudulent enlistment for this, but rather can be discharged from the Army if your unit chooses to do this. Especially if you don't have a workable family care plan after regaining the custody of the kids. Most of the time units understand that the person that has custody of someone elses children at times want to give that custody up to be able to to take better care of thier own family. That is why you will not get in trouble for fraudulent enlistment. But like I said, you do have a small chance to get discharged if you cannot provide proof that you have short and long term family care.
2- Yes, your children will be eligible for the benefits if they are in his custody. Even if you are the non custodial parent, the kids will still be considered "dependents" and you will be able to put them on the medical coverage and all that. You just will not be able to get BAH and all that good stuff unless you were the custodial parent.
3- No, the children will not be eligible for housing if you are not the custodial parent. If you give custody to the childrens father and you are not goin in as an officer, you will have to live in the barracks when you are at your duty station. Your unit can give you permission to live off base and collect BAH, but that is HIGHLY unlikely that they will do that. They will more than likely make you live in the barracks until you are at least the rank of E-5 (Sergeant), or maybe even E-4 if they have too many people in the barracks. But no, you will not be able to get housing for them. Not even on base housing. This you would have to be high enough rank to move out of the barracks or have custody of the kids or married to someone.
Hope this helps. I know it doesn't sound like stuff that you want to hear, especially when you are wanting to enlist, but it is the truth of how things work.
Good luck
