For constant angular velocity ω, g-force is proportional to the radius r: ##a=\omega^2 r##
Your "height" difference would be like a steep mountain, and air would be very thin at the small end.
The Coriolis force will give some deviation from a straight line - but you should reduce the "height" difference to get nicely flowing rivers.
"Down", towards the larger end.
Not the difference, the actual acceleration does this.
If you manage to get wind in some way, maybe. Otherwise, the whole air will follow the rotation of the cylinder without significant effects.
Only if you run at the velocity of the structure (as seen from outside), which is not likely for such a big cylinder. You would fall downwards, with some sidewards motion if the height is so large that it will kill you anyway (this happens on earth, too - for a free fall of ~100m, the deflection is of the order of 1cm if I remember correctly).
This is true on the inside, too. See my first paragraph how g-force changes with height.