Yes. https://aws.amazon.com/gamelift/

Doesn't look like it at the moment, though I imagine the .Net SDK will have support sooner rather than later. Hopefully they'll wrap GameLift into the unity sdk shortly thereafter, but to be honest I don't know where the difficulty lies in just using the regular .Net SDK.

You may be interested in SpatialOS (http://improbable.io). Disclaimer: I work there. Feel free to contact me directly if you need more info :)