If an airplane with two propellers has both propellers with the same rotation direction (say, both the propellers rotate counter-clockwise as seen from nose to the tail), will be the four left-turning tendencies still be generated?
Yes, for the most part.
Definitely P-factor will still exist and provide a yaw torque in the expected direction. This is why twin-engined aircraft have a "critical engine".
Engine torque will still produce a roll torque in the expected direction.
Gyroscopic precession will still produce a yaw torque in the expected direction when the aircraft pitches.
The yaw torque from the spiralling slipsteam interacting with the vertical fin-- which is normally quite significant in conventional single-engine aircraft-- is the one effect that may be be greatly reduced in a twin-engine airplane of the usual configuration, since the fin is not located in a strong part of the circulation behind either prop.