Why don’t hats seem to appear on Stack Exchange's flairs?
Are they OK to appear or not? I did not try myself and saw other's. By flair I refer to the icon Stack Exchange provides to put on other websites.
That won't happen because Hats require JavaScript to be added to each user's profile image.
Flair is a static image that gets scheduled for regeneration after a reputation event. It wouldn't make much sense to add it there, let alone the confusion this creates on sites that show your flair, but are unaware what Winter Bash is.
No, hats don't go on flair and don't expect that ever to be the case.