How do I become a professional game designer?
An assortment of advice:

Don't forget the Board Game Designers Forum.  

Attend Protospiel, or Protospiel South, or Protospiel North or West.

Play lots of games, and a wide variety of games, both modern and classic.

Start designing games, taking them all the way from concept to completion.  Do it over and over again.  I designed and released one free print-and-play game each month at Invisible City Productions.  I did this from January 2000 to December 2010.  The first seven games were pretty hard, but it got easier after that.

Learn how to playtest your games, and how to integrate feedback into your designs.

Attend the GAMA Trade Show.  Talk to publishers - especially smaller ones.  Take them out to dinner and ask them about the industry.

Brush up on your math – especially probability and statistics.  Learn how to mathematically  model dice pools, bags of beads, and decks of cards.