On Airbus aircraft, can the pilot control the aircraft when the autopilot is active?
Moving the flight controls above a certain threshold will disconnect the autopilot.
From the A320 FCOM (1.22.30 Auto Flight - Flight Guidance, emphasis mine):

AP DISENGAGEMENT
AP1 or 2 disengages when:

The pilot presses the takeover pushbutton on the sidestick.
The pilot presses the corresponding AP pushbutton on the FCU.
The pilot pushes on the sidestick harder than a certain threshold or moves on the rudder pedals above a threshold.
The pilot moves the pitch trim wheel beyond a certain threshold.
The other AP is engaged, except when localizer/glideslope modes are armed or engaged, or when the rollout or go-around mode is engaged.
Both thrust levers are set above the MCT detent and the aircraft is on the ground.
The aircraft reaches the MDA-50 feet (MDH-50 feet), or 400 feet AGL if not MDA/MDH, with APPR mode engaged and a non-ILS approach selected.
One of the engagement conditions is lost.

Furthermore, in normal law with all protections available, the AP will disconnect if:

High speed protection is active;
Angle-of-attack protection is active (α prot +1° is reached);
Pitch attitude exceeds 25° up, or 13° down, or bank angle exceeds 45°;
A rudder pedal deflection is more than 10° out of trim.


The different control laws act in the F/CTL COMPUTER in the following diagram:

(source: FCOM 1.27.10 Flight Controls - Description)
They are independent of the autopilot and the protections are available regardless of whether or not the autopilot is engaged.