Does the Dragon's Egg remain attached once I gain control of the character?
If so, can I now ambush Targ characters that do not have the ambush keyword, even though the Dragon's Egg is not equipped to a character controlled by an opposing player?
Unfortunately the dragon egg will fall off the character as soon as you take control of it. The condition of attach to a card controlled by an opponent will no longer be met at which point the attachment now has no valid target and becomes moribund.
The FAQ addresses this directly
Yep. As Underworld says, that one is fairly straightforward since the FAQ answers it directly.
Well I was looking for a way to keep the character with the Dragon's Egg alive....otherwise they just kill the char as part of Military claim. My only other choice may be to use Dragon's Fear and recur it over and over with Danny's Chambers. Too much effort I fear.
You can always attach the dragon egg to a valuable character that they dont want to kill?
I think it (Darogn Egg) is far more useful on locations and attachments anyway - especially if you are drawing cards during the challenge phase - there's normally a couple of safe attachments or locations which aren't going to leave play in a hurry