There's a Meta.se thread tracking the Winter Bash 2014 secret hats.
If your hat is there, then it should tell you what it (probably) means. If not, you could join the discussion there to try to work out what it means.
Alas no; not within the SE interface, anyway. However, in an answer to the question I just linked to, Aarthi (a former SE employee) mentions the one way you can, sort of, do this:
That said, there's a fairly easy "cheat" around this -- just screencap your gravatar wearing a hat, make that image your gravatar, and equip a different hat. repeat until you ...
Back in 2014 and 2015, there was one ...
See Winter Bash 2014 Secret Hats Sadly, the Time Lord hat didn't look like an actual Time Lord's hat, as your suggestion, but rather like a Time Lord mode of transport:
Time Lord was earned by editing five questions that were posted more than a year ago.
The particular trigger for this hat was criticised, as it ...
I'm guessing that there may be a hidden requirement of non-zero (or higher) scores, as theorized here: Bill Lumbergh - all 5 questions in a single Saturday?.
One of those 5 answers is at 0 upvotes, one at 1 upvote.
I'll test that theory by posting an awesome (or at least non-sucky) answer next Saturday and seeing if the very first upvote gets me the hat.