Does Bitcoin Core know how to prune the existing database or will it start the entire sync process over?
It knows what to do and you will experience no resync or reindex when you do the pruning process (barring any unclean shutdowns or sudden power loss). There is no noticeable effect when pruning except that the size taken up on disk is much smaller.
How much free disk space does the process require?
No additional space is required. Disk space will be freed up by pruning, not consumed.
Will the pruned .dat file be deleted by the process?
Yes, that is how it frees disk space.