First, I hope your daughter is alright...a week in the hospital sounds serious.
You're getting good advice from Valeska19 & KitCat. It was your daughter who mentioned ExW's feelings about your relationship with her family, not ExW. You would've been better served to listen and validate with daughter and not reach out to ExW about it unless she broached the subject. For one, ExW will likely tell daughter not to talk to you about those things and it may erode some of daughter's trust with you. For another, you're the one initiating the confrontation w/ExW which isn't going to help with your co-parenting relationship. I concur the Google Calendar doesn't seem tenable over time considering kids. I've been handling most things over email fairly successfully, but a few things here and there on text.
You mention ExW moving out of state and the affair as well in the context of a text exchange about visiting her cousin. To me that shows anger spilling into other interactions. Your anger is certainly understandable considering her actions and betrayal, but if you can process it outside your interactions and keep things to business and the kids, it might help.
Hang in there Drh. I'm sure with your Ex leaving the state and you having the kids most of the school week has to be tough, but keep being a great dad to them.
Me:39 Ex-W:37 M:7 T: 9 S:6 D:3 BD/IHS/Confirm EA/PA: Feb '20 OM1 affair ends: May '20 W/OM2 & moves out: June-July '20 W files for D: Jul20 OM2 confirmed: 9/2020 Divorced: May '21