Here's the Disneyland Today twitter account https://twitter.com/DisneylandToday Sometimes they'll tweet out a location in the park and if you're one of the first to find them, you get a small prize.
There's also a few different Disneyland apps out there that will list waittimes and daily schedules, but I'm not too familiar with them.
No on a consistent basis. Disneyland has multiple Facebook and Twitter accounts and they use them to announce a variety of things. They'll consistently post the park hours for the day and then various things throughout the day. Sometimes it's how long a line is, sometimes it's something that is happening, etc. It's pretty much never quite the same thing multiple days in a row, so you can't really count on them posting when a character is out and about.
The PP mentioned the give aways that were part of the #GetHappier Monday promotion. The give aways won't happen every Monday (at least that's what they've said when they announced the promo), but it was fun to see what they did this Monday.
For Facebook, I would look up "Disneyland Today". For Twitter, I would look up "Disneyland Today" and "Disneyland Resort". There are also more general accounts (Disney Parks Blog, Disney Parks, etc.) that are also official Disney FB/Twitter feeds.
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