Hi Cadbury:
That's what I find the hardest too....the shockingness of the lies. They floor me and I feel totally at a loss of what to say in situations like that. I'm stunned! Tongue-tied! Feel like a complete idiot afterward too!
It does help me to try to plan ahead, if I must have contact with such people. I have to psych myself up, have a reserve of responses and be prepared to exit, if necessary.
In the case of insane accusations about your parenting abilities or personal life...you might respond simply......"untrue, more manipulation", or "Not at all true. I can easily prove otherwise." or "Where are you getting your facts?" or "where do you come up with this baloney?" or "Oh come on, have you gone off your medications again???" (just kidding with that last one....

..but you get the point I bet?).
In other words be calm, try not to let the shock rule, ignor the fear, speak as if you are certain of yourself and your postitionl. You know it isn't true. You know he's a lunatic rattling off insane accusations. Shake it all off as if you are quite sure he is a nutbar. Don't let him rock your boat (plan on releasing all of that later, in a safe place, preferably with a friend/support person).
Re the picture....honestly....you have no obligation whatsoever to provide him with a single thing he requests. This jerk is recking your life, fighting you in a court of law, making up and telling authorities lies about you, acting like a complete dope and then has the idaucity to request stuff from you???????
Personally, I would not respond at all or, if I felt I must....I might say something like: "Unfortunately, I'm not prepared to do that".
It's not being petty to refuse to do favours for those who give nothing back (in his case....take, take, take.....and twist twist twist). That would be allowing him to use you. You have boundaries right? Let him cross and he'll keep going.
That's just what I think. You might decide I'm the dopey one!!

Sela