Some sites have opted out of hats. This decision was made by the moderators of that site (though, in most cases, the mods let the community decide).
The hat is displayed on the site wherever your avatar is, only to logged in users who have not opted out (in the case of SO, opted in) to hats.
The set of hats earned is different on each site. While http://winterba.sh/ ma list all hats earned, you can only use the hats earned on a particular site on that site.
Yep, the hat criteria is the same.
If this is false (probably), then you get to keep the hat (if you got the downvote later). I'm not sure about getting an upvote after a downvote--I read "upvoted posts" as "score>0"
Finally, the other SE sites are not "sub websites" of StackOverflow. They are sites belonging to the Stack Exchange Network, just like SO. "Sister sites of SO" is OK, "subwebsites" makes me grinch-y. :P