Hi there, @Agence_TOUTATIS 
Glad to say that you can hide unavailable dates using this code - Hide unavailable for booking dates.
Try it out and let me know if it worked 
The date doesn’t change when sliding the calendar carousel, since it should show the closest available for booking date. Also, if you click on a slot while sliding the carousel, the selected date will appear above.
Do I get it right that you’d still prefer to automatically update the info about the month when sliding the calendar?