Unix Network Programming, Volume 1: The Sockets Networking API (3rd Edition)
Unix Network Programming - Stevens 
There is a Unix domain socket-based mechanism for transferring file descriptors (such as sockets - which cannot be memory mapped, of course) between processes - using the sendmsg() system call.
You can find more in Stevens (as mentioned by Curt Sampson), and also at Wikipedia.
You can find a much more recent question with working code at Sending file descriptor by Linux socket.
I suggest reading this post and then deciding for yourself based on your own knowledge of your code whether or not it's OK to ignore the ECONNRESET. It sounds like your Node app may be trying to write to the closed connection (heartbeats being sent?). Proper closing of the connection from your C# app would probably take care of this, but I have no knowledge of C#.
You may have a problem if you get lots of users and if ECONNRESET causes the connection to go into TIME_WAIT. That will tie up the port for 1-2 minutes. You would use netstat on Linux to look for that but I'm sure there is an equivalent Windows app.
If you really want to get into the nitty gritty of socket communications I suggest the excellent Unix Network Programming, by Stevens.
Linux is the most accessible and has the most mature desktop functionality. BSD (in its various flavours) has less userspace baggage and would be easier to understand at a fundamental level. In this regard it is more like a traditional Unix than a modern Linux distribution. Some might view this as a good thing (and from certain perspectives it is) but will be more alien to someone familiar with Windows.
The main desktop distributions are Ubuntu and Fedora. These are both capable systems but differ somewhat in their userspace architecture The tooling for the desktop environment and default configuration for system security works a bit differently on Ubuntu than it does on most other Linux or Unix flavours but this is of little relevance to development. From a user perspective either of these would be a good start.
From a the perspective of a developer, all modern flavours of Unix and Linux are very similar and share essentially the same developer tool chain. If you want to learn about the system from a programmer's perspective there is relatively little to choose.
Most unix programming can be accomplished quite effectively with a programmer's editor such as vim or emacs, both of which come in text mode and windowing flavours. These editors are very powerful and have rather quirky user interfaces - the user interfaces are ususual but contribute significantly to the power of the tools. If you are not comfortable with these tools, this posting discusses several other editors that offer a user experience closer to common Windows tooling.
There are several IDEs such as Eclipse that might be of more interest to someone coming off Windows/Visual Studio.
Some postings on Stackoverflow that discuss linux/unix resources are:
What are good linux-unix books for an advancing user
What are some good resources for learning C beyond K&R
Resources for learning C program design
If you have the time and want to do a real tour of the nuts and bolts Linux From Scratch is a tutorial that goes through building a linux installation by hand. This is quite a good way to learn in depth.
For programming, get a feel for C/unix from K&R and some of the resources mentioned in the questions linked above. The equivalent of Petzold, Prosise and Richter in the Unix world are W Richard Stevens' Advanced Programming in the Unix Environment and Unix Network Programming vol. 1 and 2.
Learning one of the dynamic languages such as Perl or Python if you are not already familiar with these is also a useful thing to do. As a bonus you can get good Windows ports of both the above from Activestate which means that these skills are useful on both platforms.
If you're into C++ take a look at QT. This is arguably the best cross-platform GUI toolkit on the market and (again) has the benefit of a skill set and tool chain that is transferrable back into Windows. There are also several good books on the subject and (as a bonus) it also works well with Python.
Finally, Cygwin is a unix emulation layer that runs on Windows and gives substantially unix-like environment. Architecturally, Cygwin is a port of glibc and the crt (the GNU tool chain's base libraries) as an adaptor on top of Win32. This emulation layer makes it easy to port unix/linux apps onto Cygwin. The platform comes with a pretty complete set of software - essentially a full linux distribution hosted on a Windows kernel. It allows you to work in a unix-like way on Windows without having to maintain a separate operating system installations. If you don't want to run VMs, multiple boots or multiple PCs it may be a way of easing into unix.
Edited because I had to leave a meeting when I originally submitted this, but wanted to complete the information
Half of that material is learning about development in a Unix-like environment, and for that, I'd recommend a book since it's tougher to filter out useful information from the start.
I'd urge you to go to a bookstore and browse through these books:
Unix/Linux System Administration - This book covers the more system administrator side of stuff, like directory structure of most Unix and Linux file systems (Linux distributions are more diverse than their Unix-named counterparts in how they might structure their file system)
Other information accessible online:
GCC Online Manual - the comprehensive GNU GCC documentation
Additionally, you will want to learn about ldd, which is like dependency walker in Windows. It lists a target binary's dependencies, if it has any.
And for Debugging, check out this StackOverflow thread which talks about a well written GDB tutorial and also links to an IBM guide.
Do yourself a favor: go read at least first 6 chapters of this Steven's book in which you can find plenty of simple examples and many advices how to avoid common pitfalls with network programming. Without doing that you will end with a buggy, slow and incomplete client.
Your question requires more than just a stack overflow question. You can find good ideas in these book:
Basically what you're trying to do is a reactor. You can find open source library implementing this pattern. For instance:
If you want yout handler to have the possibility to do more processing you could give them a reference to your TCPServer and a way to register a socket for the following events:
So that the handler can implement all kinds of protocols half-duplex or full-duplex:
This leads to some problems:
As you see this is no simple problem. You may want to reduce the genericity of your framework to simplify its design. (for instance handling only half-duplex protocols like simple HTTP)