This appears to be Space Dandy (All quotes below taken from Wikipedia).
The central character was a bounty hunter who located illegal aliens and brought them into some central processing station for meager rewards.
Space Dandy is an easygoing and forgetful young man working as an alien hunter whose job is to discover new alien life forms across the galaxy and have them registered with the Space Alien Registration Center.
He had a robot companion for comic relief.
QT: A member of Dandy's crew, a robot that resembles and also operates as a vacuum cleaner. QT boasts being more intelligent and competent than its fellow crewmates, but its usefulness is stymied by problems caused by its outdated hardware such as memory shortages and battery outages.
He dressed and acted like a crude Elvis/Fonzie character.
Looks about right.
He hung around in a 'Hooters' style space bar.
Dandy regularly visits an intergalactic Hooters-type restaurant known as "BooBies" (ブービーズ Būbīzu) where he indulges in his posterior fetish and dreams of someday buying out the franchise.
He flew around in a spaceship. A recurring 'bad guy' followed him around in a big ominous spacecraft but never really caught him.
A gorilla-like scientist working for the Gogol Empire, Dr. Gel endlessly pursues Dandy from his spaceship, which resembles the head of the Statue of Liberty with a ball gag in her mouth that can attach to a robot body. However, despite getting close to capturing Dandy on some occasions, Gel always ends up failing in a humiliating fashion or going down with his ship.
There was an episode with a race through space and another with trains where he saved a little girl.
[Summary of episode 7:] Hoping to impress the ladies at BooBies, the Aloha Oe crew enters an intergalactic nebula grand prix against its glamorous reigning champion named Prince, his state-of-the-art robot named Z and his rat-like alien lawyer named Squeak.
I think there were only about 10 episodes produced.
There were two seasons, each 13 episodes long.