Sorry to hear about this, friend, but you're not the first and won't be the last, so, if you can, take some comfort knowing that others have managed to move on and have happy lives with good relationships with his/her kids and exes.
As for your questions, well, they have largely been covered already. However, without knowing fine details it's hard to give more specific advice, though I will try my best.
My first suggestion would be to not leave the house fully until you are both sure there is nothing to be done to save the relationship. If you immediately start house hunting, that gives the signal that you have given up, you don't want that. What you need is to talk to your partner about the situation and find if there is any chance of a reconciliation, either alone or by going to counselling. Couples having troubles is not uncommon, far from it, and many of them find it difficult to adjust, but the problems always boil down to one thing, communication. That is not to say that every problem can be fixed and that every relationship can be saved, but often, the difference between a happy and unhappy relationship can be the presence or lack of good communication.
Call your partner, tell her you want to talk about the situation, and agree a time where you can do that, uninterrupted, maybe by having a friend or relative look after the children whilst you talk if at all possible.
When you are finally talking, it is important to remember that there is a reason or reasons why this has happened, and it will be because, for some reason, your partner is unhappy. That is not to say it is all your fault, not at all, but right now, when you finally talk, you need to ask her to be open and honest about everything, and to listen to what she has to say without interrupting or getting angry. Anger is likely to happen during this, so it's important you keep it in check as anger only breeds more anger, and what you need right now is reason and positive communication.
Listen to what she has to say, take it on board, then offer your opinions on the subject, do it calmly and politely. Ask her to grant you the same freedom to speak as you allowed her, as this means you will both have had time to speak completely, and to be heard. Bad communication usually stems from one or both partner(s) not paying attention to the other. There is every chance your partner has been upset about something for a while and you've never noticed because you haven't listened to her or paid enough attention when she needed it. Again, not to say this IS the case, it's just an example.
Bottom line is to get talking, respect each other, listen to what each other has to say before speaking and most of all remember that you are not in this to win an argument or defend yourself against everything she says, you're in this to learn about what is upsetting your partner to see if you can find a way to fix it and move on. It CAN happen, you just need to be patient and respectful toward one another, and give each other time and attention to set you on the way.