Howdy! I will not be at GenCon this year, but I've been in the past.
Where can I just sit down and play games for free?
Unfortunately, GenCon is probably the worst con for this because it's so busy, crowded, and commercial. Keep in mind that there are SIXTY THOUSAND people attending, and probably 35000-40000 on Saturday alone. If this is your main goal, I would recommend literally any other convention. That said: In the past, both Rio Grande and Mayfair have sponsored free-play rooms where games from those publishers were available for demo, often with people there to help teach.
A lot of people play the day's purchases with friends in hotel lobbies, especially in the evenings. If you're going with friends (do this if you can), this can be you! If you're extroverted and able to make friends quickly (I'm not), you might be able to get in on another group's.
Where, outside of tournaments, do I need to spend the generic $2 tickets?
Any scheduled in-the-program game or seminar that isn't full will take generics. People who have registered explicitly get priority, of course, but many don't fill up.
Does GenCon take place outside of the exhibit hall?
Very much yes. It takes up the entire convention center and spills into neighboring hotels. All of those games, panels, seminars in the program... those take place in other rooms in the conference center and nearby hotels. (Nearby businesses often have welcome-to-the-con promos. They're also VERY busy because there are tens of thousands of extra people in downtown Indy at unusual times. Bring snacks (granola bars, etc) and bottled drinks.
What can I do at the average publisher's booth?
You can usually buy games, at least their newest ones. You'll usually pay list price or close to it--the only reliable way I've ever seen to get a discount is an educator badge--but you'll get whatever promos are available. You can usually get a demo and sometimes a full play of newer board games that the publisher wants to show off. At smaller companies' booths you can sometimes chat with the designers or other employees.
A lot of publishers premiere games at GenCon or have sneak previews. So you'll be going for the event, the spectacle, and being the first one to see some new things, not to play a huge variety of games.