To put things in order -
Flyweight governors have inherent droop - this gives stability to stop it hunting incidentally.
Droop occurs because the spring tension is non-linear and when it is back "on speed" to equal the spring tension it will be in a different place to equal the spring tension = droop. Google "mechanical governor droop"
If you think about it if the tension was linear it wouldn't work.
The linkage from the collective is the "droop compensation" to adjust the new set point with the change in power back to where it should be.
To adjust for more rapid changes the droop compensation is normally "over compensated".
Set 100% N2 in a Bell 206 flat pitch - pull it up into the hover and it should be higher.
The beep trim is there to compensate for atmospherics - the P in DP N1/N2 is a pressure which varies from day to day.
Set the correct N2 with the beep trim and forget about it - if you are constantly playing with the beep trim something is broken or out of rig.
The Bendix system has been around since Pontius was a Pilate and works pretty well considering. The operation and maintenance manual is around online for free.