What are the general post flight sequence checks and do they start automatically? If so, when?
All checklists are initiated by either the pilot flying (PF) when between takeoff and touchdown, and the captain (CA) on the ground.  In addition to checklists is the concept of flows, which are a muscle memory sequence of buttons to press / switches to flip / things to touch that are performed by the appropriate pilot before the checklist is called for.  The flows will happen automatically when the pilot responsible knows they should happen and the checklist will be called for when the responsible pilot is ready for it.

For example, when landing, the PF will perform the landing, the deceleration and sometimes the high speed turnoff before control is transferred to the CA for taxi.  Once the CA is taxiing and the airplane is cleared of the active runway, the FO will automatically start the after landing flow and call ground control.  Once taxi is underway, the CA will call for the after landing checklist.  

Once parked at the gate, both pilots will do their portion of the parking flow and then the CA will call for the parking checklist.

Finally, if no crew is waiting to receive the aircraft and you are not keeping the aircraft then the FO will perform the postflight inspection and the pilots will perform the terminating checklist.  If you are keeping the plane, the FO instead does a combined post/preflight inspection and begin normal pre-flight actions.

While these actions are all started by humans, the closest thing to automatic actions are the flows.  To summarize the above, with general criteria for starting flows:


Landed, clear of the active runway, CA has the plane -- perform after landing flow
Parked at the gate -- perform parking flow.


Each of these flows are backed up by checklists called for the CA.