Question

Why pressure has intensive property and is a state function?

Answer 1

Intensive properties have an "intensity" in a point in space. Extensive properties are undefined (or zero) in a point in space.

I may point to a certain point, say 100 meters below the surface of the ocean, and the question, "what is the pressure there?", makes sense. Hence, pressure is an intensive property. On the other hand, if I ask: "what is the volume there?", there would be no clear answer: I would have to specify an actual volume, and cannot make do with just a point.

Dividing an extensive property by another extensive property, say, dividing the mass by the volume, turns it into an intensive one.
A state function is a property whose value does not depend on the path taken to reach that specific value. The pressure is also a state function when system chages pressure/temperature work/heat can be observed.