So neither of his parents have taken him to the dentist or provide adequate care for him.....then why would you be in a relationship with this man?
If my exh girlfriend called enquiring about my kids dentist appointments I wouldn't amused or give her any information at all. And if I got the impression she was judging my parenting whole happily snuggling down with other parents, judgment free I can't say I would want much engagment with her either.
The picking up from school is tricky. It's his days, not her choice who does it. But I am intrigued as to why you will be doing it everytime. You are being position, by your dp (who according to you is a fairly shit parent) as the default parent. This will bite you on the arse in years to come, when the care for this child is all your responsibility.
I assume you work? What if you have to change hours? What if you want to make plans and can't pick him up? What was your partner going to do when he wanted 50:50 care but wasn't with anyone? Or with someone who couldn't do this because of work?
For you own sake, tying yourself to a man who is a poor parent (by your own description) is shuffling the responsibility to you, who is in conflict with his ex, is such a bad idea.
You think you need to do certain basic things with him, because his attention don't