I do not think it is possible to associate timers with flags. Timers are associated with sending/receiving segments actions, not flags, used in them.
Here is a summary. A more complete summary of timers is available here.
re-transmission timer is associated with a segment in the peer's sender buffer (each segment has its own timer). Once the segment is sent, the timer is started. If the timer expires before ACK is received, segment is retransmitted. No special flags are used.
persistent timer is used for so called "zero-window probes". If the receiver has full receive buffer, it sends back a packet with its window set to zero. This means that the sender cannot send any future packets.
In this case, sender will periodically send empty segments to the receiver (actually the packet has one byte in it, but it is expected to be ignored). The goal is that the receiver sends back its window and the sender learns whether the window is still zero. Note, that usually sender will learn of the window size from ACKs for other packets. This is necessary, since ACKs can be lost.
Zero window probes do not use special flags. More on zero windows can be found here
is used to send keep-alives on connections from which no data was received in a while. more on keep-alives can be found here.
This timer is started when the peer send an ACK for the second FIN during connection termination. The peer is supposed to preserve data for the connection for the value of this timer (~2mins), in case ACK is lost and should be retransmitted.
When this timer expires, the connection state can be deleted. No packets are sent.