Question About HD Video

From what I've read, it seems that I cannot post an HD video that will automatically default to HD for facebook users. The users have to select an HD option for the video. Is this still correct?

However, I still see videos playing that are defaulted to HD while others are SD. Is there a way to post HD videos that automatically display as HD instead of low res SD?

Someone suggested posting a HD video to a fan page defaults to HD quality. Is that true?

I don't think there is any restrictions about posting HD videos.
If you have any problem then, you can use Youtube, Dailymotion or any other site links to post on Facebook easily.