I saw that other SE sites have a Twitter account and announce questions or high voted answers. Should we do the same with Tor.SE? If yes, what should be twittered?
This is already happening. Every Stack Exchange site gets its own Twitter account, such as StackTor. At the time of this writing, the StackTor Twitter account has a mere two followers, so it's reach is quite minuscule, to say the least.
How it works
After a question is posted, a background process/automated bot tweets them to the Twitter account. This generally happens a few hours after a question is posted to try and get more attention from people who can post an answer or who are interested in the problem.
However, bots are pretty boring. If you follow this Twitter account, all it really is is an aggregator. Most of the accounts with the highest number of followers are people since people are generally more interesting than aggregator bots.
Thus, the best way to share questions is to do so yourself, using the "share" links found at the bottom of each question. By sharing with your own followers, people who know and trust you, you'll more likely gain clicks than some mindless bot will.
I regularly share interesting questions on my own Twitter feed, as well as Google Plus and Facebook, depending on the audience I'm targeting.