The 1st question is if a boarding happens and both ships survive it do you continue fighting boarding attacks and duels between the captains in subsequent turns until one of the ships is destroyed?
Yes, or until someone escapes jeopardy.
The second question is if one of the captains is killed but there are surviving crew for each ship, what happens during the next round of boarding attacks? That is, after you resolve the boarding battle with the 2 crews, what is the effect in the battle of one ship having a captain and the other without the captain? Does the surviving captain not participate any further?
Correct - there is no duel if only one side has a captain (just like if you're doing a "boarding action" with a cog or a sea monster.)