Yes the water pump does run of the timing belt system.
Directly from the service manual:
The 3.5L water pump has a die cast aluminum
housing and a plastic swept vane impeller. It bolts
directly to the right rear timing belt cover using an
O-ring for sealing. The water pump is
driven by the engine timing belt.
The water pump on all models can be replaced
without discharging the air conditioning system.
WARNING: DO NOT REMOVE PRESSURE CAP
WITH THE SYSTEM HOT AND UNDER PRESSURE
BECAUSE SERIOUS BURNS FROM COOLANT CAN
NOTE: It is normal for the water pump to weep a
small amount of coolant from the weep hole (black
stain on water pump body). Do not replace the
water pump if this condition exists. Replace the
water pump if a heavy deposit or a steady flow of
engine coolant is evident on water pump body from
the weep hole (shaft seal failure). Be sure to perform
a thorough analysis before replacing water
(1) Drain cooling system (Refer to 7 - COOLING -
(2) Remove accessory drive belts (Refer to 7 -
COOLING/ACCESSORY DRIVE/DRIVE BELTS -
NOTE: The water pump is driven by the timing belt.
(3) Remove engine timing belt (Refer to 9 -
ENGINE/VALVE TIMING/TIMING BELT AND
SPROCKETS - REMOVAL).
(4) Remove water pump mounting bolts (Fig. 31).
Note position of longer bolt for proper re-installation.
(5) Remove water pump body from engine
Here is link to get a factory service manual relatively cheap. 2004 Chrysler 300M Service Manual
Covers every serviceable part.