Borderlands – When can I farm a boss

Not only do most bosses respawn, even the final boss respawns.

However, the boss listed in the "Midgame spoiler" section below is replaced by a similar, new boss. And the second-to-last boss is the only other boss that doesn't respawn. Every other boss respawns 100% of the time.

Midgame spoiler:

Bloodwing does not respawn, but Son of Mothrakk spawns in her place, the fight is similar but he doesn't change elements and is generally much easier. Presumably he drops similar items.

Final battle spoiler:

Handsome Jack does not respawn, but the Warrior does. Seems you have one chance per playthrough to get Jack's Mask for this reason. You also won't get the generic loot that Jack would have dropped if you fight Warrior again, but Warrior drops plenty on his own If you get two Jack's Masks, keep one for a friend.

Any boss that drops unique loot respawns — except for the final battle (see the spoiler part above), but that boss can only possibly drop one unique item anyway: a specific skin customization.

No, Just like many other quests in the game, there are two possible rewards and you have to pick which one to go with while locking out the other one for that playthrough.

The only difference is this quest doesn't say return x to person a for one reward or person b for a different reward. You just have to make a choice and that's the choice you're stuck with for that playthrough.