How I can become a full- or part-time professional developing games? Are there specific courses? What specific knowledge should I be looking to learn? How do I enter the industry?
You can start here:
You'll find that there are lots of people who make there own games, and lots of examples of how to make and/or publish your games.
Check out the first pinned topic:
An assortment of advice:
Don't forget the Board Game Designers Forum.
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.