Since she is your "girlfriend", you currently have no options for her US immigration. She must be either your wife or your fiancee. Marriage, followed by the CR1 process for her and CR2 for HER son, is probably the best option for exactly the reasons you mentioned.
First of all, let's get the semantics out of the way.
It is not illegal to enter the US on the VWP and get married, I have done that too and there is no issue with that in itself. It is illegal to enter the States on the VWP, get married and not leave after that because this would be entering on the VWP with immigrant intent. So it is the 'not going back' part that makes it illegal.
I do not have the answers to all your questions, but I am sure other members can chip in..
The denied entry will not affect the CR-1 application, you will have to disclose it though.
Whether you want to go the K1 or CR-1 route depends on if you guys get married. If you want to bring your partner to the US to get married there, you file for the K-1. If you marry first, you will have to go the CR-1 route. You can get married anywhere. But I think that your choice for the CR-1 route is the sensible choice.
The child's father does not have to give up custody, but as you already stated, will have to give official consent to the mother to take the child to the States. I also think it would be a good idea for the parents to draw up a good parenting plan, so there will be no confusion there.
Good luck on your (long and frustrating ) visa journey!
