The current travels on the surface of the conductor rather than the cross-sectional area. Current density is closer to the measurable phenomenon. You might enjoy of good series in the Philosophy of science, studying the etymology of the terms used to express physical phenomenon and their historical progress in differential math.
Not usually. I know there is such a thing as the skin effect, but that applies not to DC or low frequency AC but to high frequency AC. DC current travels through the whole conductor, otherwise, instead of using big fat jumper cables, we could use a plastic hose with a thin coating of copper. I would suggest making such jumper cables and trying them out! Wear goggles since the copper will probably evaporate in a flash. Even for 60 Hz AC the skin effect is minimal. I know some transmission cable has a non-copper core, but this is for strength. Assuming 60 Hz transmission, skin effect is small. Yes, for high frequency signals, you CAN use hollow copper pipes.
You might enjoy of good series in the Philosophy of science, studying the etymology of the terms used to express physical phenomenon and their historical progress in differential math.
I had some of those courses years ago while I was majoring in physics. Teaching high school, I don't really get to stay in touch, even in the AP class, although we do uses diffeq for air resistance, and charging and discharging of capacitors. We do look at Maxwell's in terms of diffeq but really don't get deeper than that since it is an intro class.