Hey, if he moves away from you, so much the better.
OK, just looking at your list, let's see. Are your parents available to testify they took care of your son when you traveled? What about any of your colleagues--could they testify you made comments about having to call your folks to see how your son was? Is there anyone from the doctor's office who could testify that YOU brought him to appointments? You don't have to disprove every single lie, but if you can prove he's lying about ANYTHING, his credibility is SHOT.
Don't pay any attention to his smirking or his taunts. It's a head game. Ignore it. Believe me, he is trying to rattle your cage. Don't let him! You have the truth on your side, and all you have to do is calmly tell it. I'm sure your lawyer will tell you, no matter WHAT he says in court, keep your cool. You want to project truthfulness and reasonableness. If you get angry and emotional in court it will hurt you.
You will be fine.