-Technically, no gap is allowed, but you can take a leave for the old employer and work for the new employer during the leave time to avoid the gap.
2. Is it legally possible that my last day with current employer is Dec.1 and starting date with new employer is Dec.1?
-Yes, of course. You even can work for the two employers at the same time. So, you may put your last day working for the old employer on Dec. 1 or later, and work for the new employer beginning on Dec. 1.