Great questions! Thanks for asking, this is helpful for me to know what needs clarifying.
1. The intent is when you mark a Desperation or Wound that it's immediately added to that character's deck, and then the same Desperation or Wound stays with them until they unmark it / clear it. So if Barf has a his three calling cards and a wound in his deck, and then he marks a Desperation mid-duel, you immediately shuffle that Desperation (a random Diamond) into his deck, and it stays there even into the next scene if say his opponent runs away next round. But the value/rank of Desperation and Wounds comes up rarely enough that the exact card or value is relatively immaterial relative to just having the Diamond or Heart in the deck, so if Barf's deck gets disassembled between sessions or something, the important thing is that it comes back with the same calling cards and number of Diamonds/Desperation and Hearts/Wounds next time he's on screen.
2. A Heart always beats all other suits, and a Diamond always loses to all other suits.
3. The intent is indeed that other City players could grab the Bastard and send in goons, or even grab another character and enter the scene. Doing so might be socially impolite between players if other people aren't feeling it, and whoever's scene it is should really be the star of the show, but sometimes a great way to be the star is for the Bloodhound and her friend to be interrupted by goons and have to work together to defeat them (or whatever).
Happy to clarify further if you're curious about anything!