Scott,
<<Can the rapid extension of the nose  gear (as pushed by the
spring) cause a pressure problem in the "spool" valve in  the pump?>>
Yes.
The two "imbalances" are related in that the return fluid can't get out
of the way fast enough.  During extension the gear cylinders compete
with the door cylinders in trying to return fluid to the pump.  The door
cylinders loose because they, with their relatively small springs,
cannot exert the same fluid pressure as the gear+gravity+air spring.
The gas spring  on the nose gear, like gravity contributes to the
problem, in that, the gas spring together with the nose cylinder acts
like a pump.
Chris