Depends on the type of crossbow. There are some that CAN be used on horseback.
I wish I was that competent. Found it in a link in another forum post that I can't remember and it's an amazing site.
The only way to die is to be executed which lords are more likely to do if you been going on wild execution sprees like some players and get captured.
Lords can execute players and will only tend to do so against those that are dishonorable (Seen screenshots of dead PCs due to execution). Character death in battle does not seem to be working though, even with the death mark checked in the campaign options.
My understanding is the likelihood of you being executed is based on negative honor as it's a positive (or at least not negative) thing to execute those with negative honor.