If I damage Chewbacca with Boba Fett, can I capture him with Fett's ability before my opponent uses Chewie's reaction to prevent it from happening?

This also reminds me of another thing I'm not completely clear on: If a character with Protect takes the damage Fett would deal to another character, can I capture the character who used Protect?

That depends on who the active player is, since both reactions appear to trigger off the same event ("After dealing/being dealt damage..."). So if it is your turn, you react with Boba first, capturing him. If it is your opponents turn, he kills Boba first.

Answer to can you capture protecting characters, yes, provided they are characters as Bobas ability limits you.

Agreed. Both Boba Fett and Chewbacca's abilities are Reactions, so they would happen after the triggering effect IE: the damage being placed upon Chewbacca. Like Hida77 pointed out, the reaction that happens first is determined by the active player.

Make sense! Thanks!