Yes, this is a superior glue and will adhere to pressure treated wood
The bottle says it's designed for wood and wood composite, that's what I used it on and it worked great.
If you use any wood glue, it is important for the glue to be able to penetrate the wood fibers for a solid bond. If possible, rough up both contact surfaces. Applying a little water to moisten both surfaces will let you know that you've roughened the surfaces enough. If the water absorbs somewhat, then so will the glue. Clamping is always recommended.
Wood glue is meant for both surfaces to be made of wood. The glue must absorb somewhat into the wood fibers. I would not use wood glue to mount a light fixture under a wood cabinet. Search for a multi-surface adhesive.
Yes, it should work.
I do not recommended due to the bonding will not hold.
Yes, if allowed to dry properly
I have not tried that but I would probably say no, you would probably need something similar to an epoxy glue for that.