Q:

Who gets the bounty if placed on third party question, and the asker self-answers and accepts their own answer? 

If I assign bounty on other person question and that person answers his question, and accept it as well; and there is another answer with higher score (also posted during the bounty period): 
Who will get bounty and how much will it be if the bounty owner does not assign it manually? 

A:

If you read the help page carefully, this is actually quite clear:

Since the question asker is not the one posting the bounty, acceptance is not taken into account. 
If the user who posts the bounty doesn't award the bounty explicitly, the highest scored answer (with a minimum of score 2) posted during the bounty period will automatically be awarded half the bounty amount.