Can ASP.NET MVC application somehow interact (for example synchronize account data) with websites written on top of NODE.JS and hosted on some of 'node-clouds'?
Yes you can.
Jon Galloway has an article on this exact topic.
This is a general problem. If you have two applications written in two different languages or running on two different machines or whatever the reason is that makes them separate they can still communicate.
You fire one server on one machine and the other on another. Servers are meant to serve things, so their main functionality is to listen on a port and respond to incoming requests. But they are not limited to this. They can fire requests as well! There is no reason why server can't be a client at the same time.
Basically what you need to do is to add client functionality to the server, so it can make requests to other servers. Doing this in the context of web development is a lot easier, because we already have well known REST ( and most languages have libraries for making and parsing REST requests/responses ).
Also keep in mind that you need to implement some security features when doing this. For example when your Node.JS server is serving something only for your ASP.NET server you might want to restrict this Node.JS server, so it won't server this content to entire internet.