What effects to controllability are there when the rear doors are opened on a KC-130?
Answer based on information found on-line. Related documentation:


USAF C-130 Operations Procedures
C-130T Flight Manual
Lockheed News Service, 1957
KC-130 Specifications
KC-130J Specifications
C-130 HALO Practices



C-130J Hercules
Source: Flight Global

The KC-130 Hercules is the tanker variant of the C-130 Hercules. The cargo opening consists of a door, and a ramp which can be operated separately. 


The door is opened by rotating up and inwards the top panel.
The ramp is opened by rotating down and outwards the lower panel



Source: CQ Planes Spotting

KC-130 / C-130 have a good maneuverability at low speed. 

From the the C-130 flight manual:


Door and loading ramp are operated at a speed of 150 KIAS, with flaps down at 50%.
Same conditions for flying with door and/or loading ramp open.


Low speed adjustments:


Reducing speed from 300 kt to 150 kt requires nose up trimming.
The center of gravity may be adjusted by ballasting fuel in the tanks (refueling tank and aircraft tanks).


Opening the ramp will change the aerodynamic shape of the aircraft:


The ramp will not be moved past the horizontal position (only need for full down position is to load cargo on the ground).
Moves the nose down.
Balanced by some trim accordingly.


If loads are dropped (pallets, paratroopers) or fuel is delivered, the center of gravity will roll back, with a significant nose-down tendency, and trimming will be required.

Opening the door will be less of an issue as seen on the image below:


The aircraft is near level.



Above: KC-130J Hercules (source: Military.com).
Note that the attitude is not so far from level, in spite the door is open.