Have the Mugiwara pirates ever cashed in a Pirate for a bounty reward? Is the "Bounty System" in One Piece really necessary? I mean I have never really seen anyone get the bounty reward after capturing and handing the Pirate to the Marines.

Is there any episode where someone actually claimed the money after catching some pirates? What I know is Zorro's previous occupation was "bounty hunting", but did Zorro ever get the money?

Are there any episodes in One Piece that interpret how the procedure of taking the bounties from Marines work?
Yes, Zorro received money and lived this way though this was not his intention or occupation, he simply wanted to fight strong guys.

The bounties are based on the level of threat that person is for the world government.  

Also if you have a bounty on your head and you kill or defeat another wanted pirate probably wouldn't be a good idea to take it to the marine force and ask for the prize cause you're actually turning yourself in.