The answers to your immediate problems would seem to be in a fixed schedule for the kids. She will always have some excuse why she should get what she wants. My W is the master of this. She always has an excuse why I should have the kids so she can go party. In the case of the Bday dinner, you should not give in so easily. She will not respect you if she can walk all over you. Don't let yourself get pushed to the background of your kids lives.
As hard as it is, I would find some hope for your situation in how ridiculous it is. Your W left you for your neighbour. It's like something out of a Soap Opera. It won't last. It won't.
I get the impression that you spend a lot of time feeling sorry for yourself. Do you get out and do things for yourself? do you a have a release for your sadness and frustrations? The gym works for me.
M35 W37 S9 D6 M12 yrs Know 15 yrs Bomb 1/28/07 My Sitch Failure is the opportunity to start again more intelligently - Henry Ford