TCP Session Keepalive and Idle timeout

1. the idle values are local to the BIG-IP, and are used to determine when certain entries can be removed from the connection table. they depend from TCP profile settings and -since F5 is a full proxy- can be different for client- and server- side. 
2. this behavior can change, depending on configuration. in the Virtual Server configuration, you can set up which TCP profile should be use for each connection, and in current BIGIP versions "server-side TCP profle" usually defaults to "use the same as client-side". so if you configured your VS to use the same TCP profile for both client- and server- side connection, yes they will be the same
3. yes, this is most likely a RST sent by BIG-IP to close client connection, as default idle timer for TCP profile is 300s 

the vserver tcp behavior is determined by client side and server side tcp profiles of each virtual server.

https://my.f5.com/manage/s/article/K13004262

https://techdocs.f5.com/en-us/bigip-16-1-0/big-ip-local-traffic-management-profiles-reference/protocol-profiles.html#GUID-A2C43B65-86F4-4618-91AB-169B7B46A206