Why is pressure negative in force and work
In this context we are talking about the work done by a gas.

Rather than representing it as $w = -PAD$, think of the formula as representing an area (that the gas is pressing against) moving a distance $D$. Since this a change in volume, we can set $w = -P\Delta V$

If you are putting work into the system (say pressing a piston into a cylinder containing a gas at a given pressure), the volume is becoming smaller, so $\Delta V$ is negative.  The negative of the pressure times a negative $\Delta V$ is a positive value, which matches with the sign convention for putting work in.  Vice versa, an expansion of a compressed volume at a given pressure will push the piston back out of the cylinder and allow the gas to do work, so $w$ is negative which follows the sign convention for work output.