OK, On the dental issue.....I think you responded poorly.
You explain yourself too much. The answer should have been "noted".
Then you do exactly what you said.....check it out yourself, either take the child to the appointment on your time, or reschedule at your convenience.
Maybe that was the ONLY appointment Mom could get? See....that's what she will say in court.
My son once fell and had to get stitches.....and they needed to be taken out on a certain day after they were put in. Let's say that was Day 5 to be taken out on Day 5 and that our son had an appointment on that day at WHENEVER to have them removed. Dad got pissed because HE can handle making appointments himself. See.....dad focused on CONTROL during HIS time. Whereas I focused on our son -- knowing that by the time dad would call all appointments for routine stuff would be booked -- afterall, I used the same pediatrician for all three kids and dad had no clue how far out appointments were made from the day of the phone call. The point is to focus on the child....if Dad couldn't make it at WHENEVER, then by all means, cancel and figure something out. But if something can't be figured out, dang it, I got an appointment for our son already set.
I think I've said this before -- watch the control factors and focus on the child. Don't argue unnecessarily - in my case dad looked like the fool (and you're on the receiving end of similar information on the dental issue -- so yep, you know what I'm thinking).
Make your own call, and choices, move on and inform mom at the next exchange.