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 ...
The lack of severed Wampa heads seems to be unfair, particularly to Scifi.se :(
Otherwise, I say this looks like great fun, and I think we should participate!
After all, anyone who feels that such hat-related fun is ill suited towards our serious nature here on Scifi.SE can always use the "I hate hats!" button to opt out.
Vote for which hats we want to have in: Hat suggestions 2012
I vaguely remember this from last year. Essentially, there were two types of hats, some that were generic, and others that were specific to a tag. We could make suggestions for the specific ones. I would like to propose the following:
star-wars Wookie Hat
firefly- Jayne's hat