For example, I have an endpoint that performs tasks A, B, and C upon client request. If the client sends a request, task A completed but before tasks B and C start client closed connection, network issues, closed browser windows, etc... My question is, will tasks B and C be completed or not?
Your nodejs program cannot tell whether the client connection is closed until it tries to send something to the client.
So, generally the processing of an incoming request runs to completion before it finds out the client has gone away.
This is a nuisance when a client gives up on a slow request: the work is wasted, or, worse, an update completes without any way to tell the client it is complete.