here are your answers:
The chieftain's arrow can be taken from the pool when you roll an arrow. Can Elena Fuente hand out the Chieftain's arrow?
Yes, she can!
Can a player take the Chieftain's arrow if they shoot El Gringo?
Yes, they can.
The broken Arrow roll removes an arrow, but cannot remove the Chieftain's arrow. Can Kit Carlson remove the Chieftain's arrow with his ability?
Yes, he can do it.
Can Bart Cassidy take the Chieftain's arrow instead of damage, or just regular arrows?
He can choose to take the arrow of the Indian Chief.
One more question for Elena, does she give the arrow out instead of taking it herself, or does she give the arrow after she takes it? Mainly just checking in regards to the Indian attack once arrow pool is empty.
She directly gives the arrow instead of taking it for herself.
In case she is supposed to get more than one arrow, she decides in which order the arrows are taken (i.e.: she could give one arrow to another player, trigger the indians attack, then take the remaining arrows for herself; or she could do everything in the opposite order...)
Hope this helps