Regarding the first question, IMO the webserver should half-close the socket. So, the on the wire requests are rejected on TCP level, while the final response is being delivered. Of course, the client needs to deal with socket errors in addition to http status codes.

For the keep-alive header, you are right, I wasn't aware of it.

I'm not sure what you mean by half close.

In my scenario at time N, the connection is idle -- both sides have received all data the other has sent, all requests have received a response.

If the server half-closes (through shutdown) and sends a FIN, simultaneously with the client sending a new request; that enables the server to read the request, but not respond to it, so I don't see how that is helpful?

The problem from the client side is it's sent a request, and seemingly in response the socket is closed. That could indicate the server crashed on the request, or the server closed the socket because it was idle. If you have a request that you know or suspect shouldn't be made more than once, you shouldn't retry it on a new connection. Assuming the tcp packets from the server, you can actually take a good guess at causality, because the ACK number and TCP Timestamp indicate if the server saw your last transmission, but that information isn't exposed through normal the normal socket API; you could maybe guess based on round trip time too, but it is nicer in http/2 (or other protocols), where there is an explicit close message.