In the short term you are in for some hard times, but I think that if you act consistently and lovingly over the longer term, that is what the kids will appreciate and respond to. You can't change their mother's histrionics or what she tells them when you're not there, but they will know when they feel secure and loved and all the routine stuff you do with them will help with that.
I totally agree, well said.
This will make it easier on you anyway - have your own little world with the kids regardless of what their mother does. Consistency is what these little tackers need, and they will eventually work out that their mum isn't always right about things. Don't dispute things she says to them, or 'talk to them' about it too much, just be you, same you, same routine, same smiles and safe place to be when they are with you. After dinner games, a cooking session in the afternoon (easy bikkies etc), maybe the bigger kids can tell YOU a bedtime story. Distracting them is an absolute must - especially if the older 2 are used to being in school all day. It's not supposed to be a summer camp - this should all be primarily done by the kids dad. Each time the mums tells them their dad sucks, they can think "well he wrestles REAL GOOD", or 'well he tells a killer bedtime story', and any apprehension will recede over time.
I strongly advise you leave the discipline to their dad too.

It can be very hard with a grizzy, mopey 3 yo, I'm not sure on the TT thing myself, but try to remember that when she is at her whiny worst - it's the time she needs you most of all. However if comforting her makes her worse - leave it. They are darn good at manipulating their environments so don't be a doormat!
Be firm and smiley! It will take time, hang in here.

P.S - you are right, the children will never understand if their dad walks away. I get the line of thinking, but it never works.

P.P.S - do have have to have legal representation in the court? Can DH represent himself? It's a loooot of $$$, I'm sure the ex wouldn;t be such a pain if she was paying out that kind of money.