Before we go on here....there's a very suspicious hole in this story. As someone who has listened to many stories about domestics, I have to ask. What, exactly happened between "naturally I was very upset and ask her to keep a better eye on the kids...." and "when I come home, she is gone."? Was there any sort of violence or altercation before you left the house that morning, even down to the " she struck at me so I caught her wrist" sort of wrestling? Any ultimatums given? If so, you will probably be hearing all about it, and not in a positive way when you guys get to court.
As it has been pointed out several times here, this whole scenario makes you sound like you were at the very least not doing well with dealing with a wife and four small children. This situation, even if you are simply sued for divorce, and do not have a restraining order against you, has all the hallmarks of something that will go into a terrible, long drawn out custody and visitation and pick up and who gets vacations and claims on taxes mess. Because it sounds as though you two have this dynamic going.
And the fact that you refer to your wife repeatedly as "they" means that you see yourself as fighting both her and her mother. But I'd predict that when she falls out with her mother, she'll be back with the kids, and there will be more drama, more craziness. You are, should you choose to accept this mission, going to have to provide stability and balance for four little people finding their way in the world for a lot of years in the future. No one, at this point cares whose fault it is that they got here, what who said about it, and who got fixed. They're here, and they're yours to care for.
I would very strongly advise you to talk to a good family attorney, get some counseling for yourself, which will play into any court situation magnificently, and go forward with this without the drama and the Oh I am missing my kids soooo much! What am I to do? Also, you seem to be holding your work up as the sacred standard, as in "I can't be troubled about the kids, I have to work!" While you will want to arrange custody and pick ups around your employment, you can't spend the next twenty years fussing about schedules. You will need to have something easy and possible to accomplish and get to work. Let your attorney help you with this after you get your breath here.