Is it me, or does anyone else think that the really surreal thing here is not that Anna Chapman "proposed" to Snowden on twitter but that Chapman is even on twitter in the first place?

Why not, she is now a celebrity in Russia that is hosting a TV show and works as a model. As such it would seem natural to me that she is using social media.

What is surreal to me is that a spy turned into a celebrity in the first place.

What, you mean Snowden's not a celebrity?

Or that she's proposing to someone who, last I heard, was still in a committed relationship with someone else. :P