StarCraft II seems much more spectator-friendly (and modern computer), but are there still any large tournaments that are sticking to the old, balanced to the n-th degree SC1? I can't really make out what happens in Korea that well, it seems like there are tons of StarCraft tourneys, but are they all SC2 nowadays?
Brood War is still very popular in South Korea. In fact, it's still the major eSport there and it's hard to look into the future to say if it will stay that way. Blizzard has made numerous attempts to advertise Starcraft 2 in South Korea(such as prohibiting broadcasting rights of Brood War games on an IP-rights basis, which failed), but so far, while it is going relatively good there, there's not much more going on than the GSL.
However, outside of South Korea, the competitive Brood War scene is dead. There are no professional teams anymore and also no tournaments which pay out money.
Starcraft 2 has pretty much taken over the entirety of the population of Brood War as well as Warcraft 3 (which, together with DotA, is still pretty big in China, though) outside of South Korea. All major organizations (MLG, ESL, pretty much all smaller tournament hosters) have switched to Starcraft 2 and it's only going to grow from here on out.
Balance-wise, it's hard to say. The game literally switches around every 3 months with one race dominating every tournament. That probably won't change until Heart of the Swarm comes out, which is of course going to tip the balance scale around a lot.
The answer to your question is yes. The pro-gaming circuit 'StarCraft: Brood War professional competition', which you can find out more about here, still hosts professional matches and tournaments in the original Starcraft game. However the number of professional matches in Starcraft 2 is likely to increase with the release of future Starcraft 2 games and Starcraft 2 is already featured in competitions at events like Blizz Con.