I got the hat "retro fan" and I got 2 reputation points. Once the hats are gone, do we lose the amount of reputation we get from them or do we keep it? For example, if I got the hat X and get Y reputation points, once it reaches January 4th, will I keep my reputation or lose Y reputation points? Sorry if this is a really bad question!
I realize that there are already four answers all saying basically the same thing, so a fifth answer would be kind of redundant. However, there is an important point that none of the other answers addressed. It is true that no reputation is awarded for hats; but we can go further than that. We can actually see what you got reputation for.
Right away we can see that you have only ever earned reputation on two sites — Stack Overflow and Meta Stack Exchange — so we won't have to do that much digging. Let's start with your Stack Overflow account.
If you go to the Activity page in your account and look through the tabs there, one of them is called "reputation". This tab contains a record of all the reputation you have earned on Stack Overflow, with the more recent earnings at the top. Here is a screenshot of what we find:
A quick glance is enough to tell us that you haven't earned any reputation on Stack Overflow since November 28th. Clearly the reputation that you think came from a hat cannot have been awarded on Stack Overflow, as Winter Bash had not even started at the time of your last reputation awarding.
This leaves us with your Meta Stack Exchange Account. Following the same procedure, we get this screenshot:
As we can see, all your reputation has been earned today or yesterday, and it all comes from two posts. Yesterday you earned 15 reputation (three upvotes of five reputation each) for your question about the leaderboard. Besides for that you had no reputation changes prior to posting this very question. Thus, there does not appear to be any instance of you getting a +2 reputation prior to asking this question about the +2 reputation that you got.
Now it is a little harder to track the reputation notifications that you might have received after posting this question, because you got some negative reputation. Negative reputation does not generate a notification when it is earned; instead it is held until you earn positive reputation exceeding the negative reputation and then you get the notification showing you the difference between the positive and negative reputation.
In this case you started off with three downvotes, for -6 reputation. You would not have received a reputation notification at that point. Then you got an upvote for +5 reputation, but you would still not have received a notification since the negative reputation still exceeded the positive reputation. Then you got three more downvotes for -6 reputation, which would have you at -7 reputation needing to be applied. Then you got two upvotes for 10 reputation, so you should have gotten a notification of +3 reputation (10-7=3). Then you got another downvote and another upvote, which would give you another notification of +3 reputation (5-2=3).
Based on the above, it would seem that you did not receive any +2 reputation at all during Winter Bash. The only reputation that you earned during Winter Bash prior to posting this question about whether you would lose your hat-generated reputation was the +15 that you got for your earlier question about the leaderboard.
If you want to try to figure out precisely which reputation gain was the one that you associated with getting the hat, you would have to figure out when you got the hat. First we have to check which site you earned the hat on. A perusal of your profile page shows that you earned it on Stack Overflow:
Knowing this it should be pretty simple to figure out when you got the hat — the hat you got is for voting on a post over a year old. You can check your voting history by going back to your Activity page and clicking on the last tab "votes". This tab is only visible to you so I can't provide screenshots, but once you go to that tab, you will find a bunch of sub-tabs, the second of which is "upvotes". By clicking there you will find a list of the posts you have upvoted. All you have to do is find the post that was over a year old (or if there is more than one such post, then the first one from after Winter Bash began). When you find the post you will see a number of days or a number of hours or a date listed to the left of it. If you hover your cursor over that number/date it will provide the exact timestamp of the vote. Once you have that information you can go back to the reputation tab in your Meta Stack Exchange Account and compare the timestamps there to see which reputation change occurred around the time that you cast the vote on the old post, and that will presumably be the reputation change that you thought was from getting the hat.
Any votes you cast, comments you leave, ask questions, post answers, or any of the various other actions you can do on the network, and the reputation and badges earned via the same remain even after WB is over.
The triggers for the hats depend upon doing a specific task on the network, and that's where their relationship ends. The hats are related to the task you do on the SE network, not the other way around.