These are a lot of awesome points! To elaborate a bit -
Specializing in something is always a plus! Do you want to be an animator, sound designer, QA, etc.? Having a bit of knowledge about game design as a whole is awesome, but it's also important to specialize in something (unless you know everything about game design ever. :P) Figure out what you like and learn, learn, learn. QA roles can be an awesome jumping point into the game industry, and many people learn a craft while working in QA. I have friends who began in QA and are animators now. :) Many companies have mentors that can help you reach your goals in the industry!
Game conferences are awesome. :3 I've been to GDC and it's a great event for checking out games, having your work looked at, and meeting up with recruiters. GDC is actually how I met a recruiter for Riot. :o
Getting your foot in the door is the hardest part. As An Honest Thief said, you need experience to be able to get experience in a lot of cases. Luckily, that isn't always the case! Personally, I started off at a smaller company to gain some experience, and then went on to pursue a job at Riot. Smaller companies are great - both to gain experience from, and to work at! Internships are also valid real-world experience.
Being 'in the loop' with the industry is also helpful! It gives you the DL as to what's going on, what's hot, what's up and coming, and can make for some insightful conversation with a recruiter.
Oh! You're definitely right that we all don't have degrees in Game Design; I majored in Graphic Design myself, and actually started off as a designer/writer for a local entertainment magazine. :P