....and participates in a challenge with a Dragon. What happens? Can Eddard kill the dragon with Ice and then he dies himself due to Dragon Bite?

Ok, so go ahead and open up the faq to the "Challenges Phase" flow chart,

Lets say Eddard is declared as the attacker, This is the second "frame work action" step (first step is just the start of the phase).

Then there is a "player action window" for player actions, Using ICE is a player action (you know cause of the bold "phase:" text, side note: the exception is "response:", responses are not player actions). However! since there are no other participating characters yet, Ice cannot be used yet.

Then, another Framework action window. Stealth where applicable, and defenders are declared. Say a dragon. It is after this second step within the framework action window that both characters are officially participating. you might want to look at the "framework action window" flow chart a bit more closely at this point. on step 4 of the flowchart, passive abilities resolve. In this case Dragon bite.

Basically, Eddard dies and is removed from play before you reach the next player action window after defenders are declared.

Thank You very much!