InChina:
What we are especially wondering is did we do anything against anything found in the Boeing manuals.
To stictly answer your question - Yes as a crew you did:
OM B vol 2 chapter 11:
ADVISORY MESSAGE:
DRAG REQUIRED
CAUSE:
Airspeed is 10 kts or more above FMC target speed or within 5 kts of Vmo/Vmmo.
CORRECTIVE ACTION:
Use speedbrakes, trim or reduced thrust, as required, to bring the airplane within 5 kts of FMC target speed.
Since thrust was already at idle your only corrective action available according to the manual would be to use the speedbrakes. You did
not do so accordingly.
InChina:
I then entered some descent information on the FMC. Looked up and saw a rapidly (very rapidly) increasing green speed trend arrow. Which resulted in a Mach over-speed warning.
This may account for the discrepancy between the recalled path deviation from you and the captain. If you changed e.g. an altitude constraint or descend winds, deleted a waypoint etc, the FMC calculates a new path. One of you may remember the old path and the other the new. If the new path suddenly predicts you much higher, the autopilot will increase vertical speed, maybe contributing to the pending overspeed situation.
As stated by many other posters I too would recommend to abandon VNAV at the moment DRAG REQUIRED is annunciated. This is another good reason to switch to LVL CHG when above the path and messing with the FMC, since you don't know witch immediate effect it may have on the flight path.
InChina:
As I stated we did spot the airspeed trend arrow increasing and we tried to correct accordingly. Again, our actions might not have been the best or as fast as needed, but were they actually wrong?
Before the overspeed condition, I would have pulled the wheel into CWS as some other posters also mentioned. But
in the overspeed condition Boeing recommends to leave to AP engaged.
FCTM chapter 8, Non-Normal operation: Overspeed:
When encountering an inadvertent overspeed condition, crews should leave the autopilot engaged unless it is apparent that the autopilot is not correcting the overspeed. However, if manual inputs are required, disengage the autopilot. Be aware that disengaging the autopilot to avoid or reduce the severity of an inadvertent overspeed may result in an abrupt pitch change.
During climb or descent, if VNAV or LVL CHG pitch control is not correcting the overspeed satisfactorily, switching to the V/S mode temporarily may be helpful in controlling speed. In the V/S mode, the selected vertical speed can be adjusted slightly to increase the pitch attitude to help correct the overspeed. As soon as the speed is below VMO/MMO, VNAV or LVL CHG may be re-selected.
You did the right thing in switching to V/S. As can be seen from the above the captain incorrectly switched to LVL CHG. Maybe you can go free on this one, especially since the captain also declined your request for using the speedbrakes as appropriate according to the manual in the first place. But I guess it depends on your morale and if you feel that the captain asked you to operate the aircraft differently than you yourself intended and if this was the cause of this situation.
Good luck - especially in finding a better company. You seem like you have a good attitude, do not shy from your responsibility and have willingness to learn from this experience. All good qualities for an aviator in my opinion.