I remember playing the demo of JA1 back in the day and loving it. But I couldn't afford the full game. Now I have a chance to buy it, but I'm wondering, which is considered better, 1 or 2? And why?

I'd suggest JA2, mostly due to the still active community, and the 1.13 patch. After you get your bearings with JA again, drop by the Bear's Pit to discover a whole new game.

How strongly are the two games connected to each other in terms of plot/characters/etc.?

Also, what is Jagged Alliance: Deadly Games?

I'm wondering which of the 3 ones on sale are worth buying, as I've never gotten around to trying out any of the JA games.

If you buy one, get JA2 - It's the latest, the best and as JMich said, JA2 has amazingly active community even after all these years, including the 1.13 patch.

All games are standalones. They are connected in a way that there some characters (mercenaries you hire and fight with) are the same, but you don't "lose" anything if you haven't played JA1 when playing JA2.

Deadly Games is kind of "scenario-based" or "mission based" than JA1 / JA2, which are free-forming with their gameplay. In JA1/JA2 you have an island to conquer and in JA: Deadly Games you have a storyline to follow which goes on as you do missions.

Of course, nothing wrong with getting all three if you want and have time to play :)