How fast can a large passenger jet descend and slow from cruise to emergency land?
A 5-minute descent from FL320 to 10,000 feet is pretty quick. As noted in the comments, the descent rate is 4,400 ft/min. For reference, during final approach descend rate is 300~500 ft/min in a small airplane, 500~700 ft/min on an airliner. A typical descent from cruise altitude is performed at 1200~2400 ft/min.
Getting down fast
A &quot;dive bomb&quot; flight-simulator-style descend is not practical if you intend to land. After all, you have to arrest the descend and bring the aircraft to a stable state. You may very well break the plane when trying to pull out of the dive.
Now, what is the fastest way to lose altitude? According to the Boeing 737 QRH, the Emergency Descent checklist (which is conveniently listed on the first few sections) says...

(...) Spreadbrake...FLIGHT DETENT
(!) If structural integrity is in doubt,
limit speed as much as possible and avoid high maneuvering loads.
Set target speed to Mmo/Vmo. (...)

In other words, deploy the speedbrakes, then pitch down to max airspeed.
Note that, however, this procedure is only used when it is absolutely necessary to get down as quickly as possible. The Boeing 737 FCOM also contains the following, in bold letters:

WARNING: Use of speedbrakes at speeds in excess of 320 KIAS could
result in severe vibration, which, in turn, could cause extreme damage
to the horizontal stabilizer.

Landing gear &amp; Flaps
Deploying the landing gear is not good because:

The maximum flying speed with gear extended is 320 knots, while Vmo is 330 knots. Therefore, if you descend with the gear extended, you will lose altitude slower.
The maximum flying speed for extending the landing gear is 270 knots. Therefore, if you wish to extend the gear, you will have to slow down to 270 knots first, then lower the gear handle, wait for the gears to be locked, then pick up speed again.
Flying with the gear extended introduces vibration to the airframe. Deploying speedbrakes already introduces vibration. A damaged engine further adds vibration.
You need the gears to touchdown, so better protect them in the landing gear bay.

Extending the flaps is even worse because the maximum speed for flying with flaps is even lower. The maximum flap extension altitude is also 20,000 ft.
Cross control
Cross-control can probably lose altitude even quicker, but it is not a maneuver you would do on a passenger airliner. When crossing control, the gravity vector is not down, but sideways - meaning passengers will be forced to the side of the seat, a sensation they are not used to. Furthermore, the passengers sitting on the window side will likely perceive the attitude as extreme bank angle and think that the plane is going to crash! The last thing we want in this scenario is a cabin with panic passengers.