What are the retransmission rules for TCP?

What exactly are the rules for requesting retransmission of lost data?

The receiver does not request the retransmission. The sender waits for an ACK for the byte-range sent to the client and when not received, resends the packets, after a particular interval. This is ARQ (Automatic Repeat reQuest). There are several ways in which this is implemented.

Stop-and-wait ARQ
Go-Back-N ARQ
Selective Repeat ARQ

are detailed in the RFC 3366.

At what time frequency are the retransmission requests performed?

The retransmissions-times and the number of attempts isn't enforced by the standard. It is implemented differently by different operating systems, but the methodology is fixed. (One of the ways to fingerprint OSs perhaps?)

The timeouts are measured in terms of the RTT (Round Trip Time) times. But this isn't needed very often due to Fast-retransmit which kicks in when 3 Duplicate ACKs are received.

Is there an upper bound on the number?

Yes there is. After a certain number of retries, the host is considered to be "down" and the sender gives up and tears down the TCP connection.

Is there functionality for the client to indicate to the server to forget about the whole TCP segment for which part went missing when the IP packet went missing?

The whole point is reliable communication. If you wanted the client to forget about some part, you wouldn't be using TCP in the first place. (UDP perhaps?)

There's no fixed time for retransmission. Simple implementations estimate the RTT (round-trip-time) and if no ACK to send data has been received in 2x that time then they re-send.

They then double the wait-time and re-send once more if again there is no reply. Rinse. Repeat.

More sophisticated systems make better estimates of how long it should take for the ACK as well as guesses about exactly which data has been lost.

The bottom-line is that there is no hard-and-fast rule about exactly when to retransmit. It's up to the implementation. All retransmissions are triggered solely by the sender based on lack of response from the receiver.

TCP never drops data so no, there is no way to indicate a server should forget about some segment.