Sorry to hear about this -- that's quite a surprise to see that issue suddenly cropping up. It looks from that like it's honestly a bug in the Steam API right now. Are you on the very latest version of the steam client? It's possible that they have a passing bug inside that one, or the one before it, that is leading to this. The error that you're noting is something that is coming from within the Steam system itself.
The other possibility is that it's a corrupted download of that steam_api.dll file for some reason (sometimes Steam mangles a file). Simply going into your game folder and deleting that dll and then right-clicking the game in steam and choosing properties, then hitting Local Files, and Verify Integrity Of Local Game Cache should replace that file that you deleted as well as fixing any others.
If you don't care about achievements being logged to steam, you can also just leave that deleted and it won't fire up and it will stop impacting the game at all, too, just as an aside. But I figure that you probably want your achievements logged!
You can send the full log to arcengames at gmail dot com and we can take a look at it, but I think that you got the relevant part actually. Usually there's a lot of repetition of the same thing over and over again when it's an issue like this particular one.
We're still going to be mostly out of the office between now and the 2nd, so apologies for being slow during this period. If none of the above works, you can always temporarily delete steam_api.dll and it won't log your achievements to steam, but will log them locally. Then when we get the rest of it figured out (or steam issues a new update), the updates will actually sync from your local machine up to steam, which is handy.
Sorry for the trouble there!