Just to clarify a bit. I hope this also answers your questions.
timeout http-request
Is the time from the first client byte received, until last byte sent to the client (regardless of keep alive). So if your backend is too slow or the client is sending his request too slow, the whole communication might take longer than this, and the request is dropped (and a timeout sent to the client).
timeout http-keep-alive
The time to keep a connection open between haproxy and the client (after the client response is sent out). This has nothing to do with the backend response time. This has nothing to do with the length of a single request (i.e. http-request timeout). This allows faster responses if the user requests multiple ressources (i.e. html, img, and js). With keep alive the single requests can make use of the same tcp connection. This way the load time for a full webpage is reduced.
timeout server
This is the timeout for your backend servers. When reached, haproxy replies with 504 (gateway timeout). This also has nothing to do with keep alive, as it is only about the connection between proxy and backend.