If she told you now that she plans on leaving now (before I recall you heard through word of mouth), you now advise her via email and registered letter that you do not agree with her unilateral decision to relocate with the children and that she does not have your consent to move the children.
You need to be on record and have it in writing that you don't agree with her moving with the children. If you sit on your hands and do nothing and she moves, a court may deem your inaction as consent. So you have to advise her IN WRITING.
As for the house, is it something you could carry on your own? If so, get a valuation done to figure out the fair market value so you can buy her out. As for the paying for the house and c/s, well.....that is likely going to happen regardless. If you make more money than she does, it really doesn't even matter if you get 50/50, you will be paying c/s. Full c/s if she has the kids more than 60% of the time, and offset if you have the kids more than 40% of the time.
So don't get too worked up about c/s. It is the impending move WITH THE KIDS that you have to stop.