Me and my ex husband have been separated for over a year and we have joint custody of our daughter. She resides with me. However, I informed him 3to4 months in advance that I will be going away and would need him to watch our daughter. He agreed but is now stating that he will not watch her while I am gone. I am only going away for 10 days and cannot be refunded my expenses for this trip. If I drop her off to him and leave her with him so he could watch her, is this grounds enough for him to file abandonment.

When looking for help on a site like this its always a good idea to browse around first. Had you done so you might have found this sticky:

https://www.askmehelpdesk.com/family-law/child-abandonment-read-first-364259.html

So no, this is not enough to file abandonment, but if you have full joint custody with neither a primary, it could be enough to start a custody battle over.

No, since you are arranging care of the child