Can I give away some keys I have for Origin?

You can give them away but only on the forums in whatever way you'd like.

You can't create actual site giveaways for them though, enjoy!

you should read this

http://www.steamgifts.com/discussion/GOv22/drop-your-orphan-keys-here-thread-not-sure-if-key-got-used-or-desura-other-platform-keys

But can I trade them in the trade area?

Thanks for answering.

As the others said, you can't make a standard giveaway and gain levels, but if you still want to give them away just to be nice I'm sure there are people who gladly takes an Origin key as well.

One nice thing I've seen others do is that when they make a puzzle they write the non Steam key in the description of the private giveaway you get to for solving the puzzle. That way there's a bonus for whoever is quickest to solve the puzzle, and any bots farming the forum for keys can't get to them.

Requires an additional (Steam) giveaway for the puzzle though.

Not officially here, although you can always just make a new thread and use random.org.