As an experienced .net programmer, why should I study node.js?
For example if you are interested in asynchronous programming model or you want to use single language across your client/server stack or your want to try slightly different way of doing things and so on ...
Does it replace my c# code?
In general it can both replace and complement various parts of your existing applications (not only) written in C#, however it may depend on the scenario and whether it's suitable to use patterns and paradigms which are applied with node.js.
Can node really compete with dot net (database support, multithreading, OOP, readability, etc.)?
Node.js has very active community which produced so far around 4.5K packages which covers wide range of areas, functionality and systems, for example you should be able to find several packages for every major (and even minor) database system without problems.
Do you really develop faster with node.js?
Node.js is less verbose in code quantity than C#. Area where it particulary excels is for example network programming. Its network related API is rather simple and you should be able to write these kind of programs fast. Of course it's not limited to only network stuff, for example node is extensively used in web based (real-time) applications. You should however be careful with asynchronous/callbacks stuff and learn or read some articles on this stuff before diving headlessly into node.