Why do propeller driven aircraft (especially smaller ones like fighters) tend to have a narrower wing with a higher aspect ratio and a more perpendicular wing angle in relation to the body, whereas jet powered aircraft have a broader wing with a lower aspect ratio (a more "delta" shape) and also more angled? Does it have something to do with the operating speed and aerodynamics?
Speed is the reigning factor here, and aerodynamics as well of course.
Propeller driven aeroplanes operate at lower speeds than jet engines - propellers are more efficient until compressibility effects start to appear, and it is these same compressibility effects that influence the wing shape as well. Compressibility as in: give air enough time to move out of the way and follow the wing contour. Usually, propeller aeroplanes operate in this aerodynamically docile speed regime.
A wing creates lift by deflecting air downwards, and is most efficient if there is a large amount of air deflected a bit: the least induced drag is created. So the wingspan is the most important parameter here. Lower speed aeroplanes will stick their wings straight out, as much as possible without creating large structural issues due to wing root bending loads becoming unmanageable.
Aerodynamics at higher speed.
Once shock waves start to appear, drag rises rapidly. But only the air flowing perpendicular to span concerns us for the lift creation, and therefore we can present the wing at an angle and use the lower perpendicular speed component for lift creation. But swept wings create many more structural problems than straight wings, and also since we're going faster already the argument of more air at less deflection speed is of less relevance. We have many more problems to tackle here than only minimise induced drag.
Yes it does. The reasons are many, and complicated, but I will try to briefly summarize them here in simple terms.
A jet-powered fighter has to pop through the sound barrier with minimum ill effect and be aerodynamically predictable at supersonic speeds. It also has to withstand the great stresses associated with these actions and those imposed by high-G maneuvering, without failure. All these thing argue in favor of stubby and sharply swept wings that are either trapezoidal or delta-shaped.
I invite the experts here to add to this list and to perhaps offer more technical background.