Without air resistance, bodies travel in a parabola. What is the curve that is travelled when air resistance is included? I found that there are different air resistances, which curves would these types produce?
With air resistance, you don't get a nice curve (such as a simple parabola). I don't think there is a special name for all those types of curves you can get.
For air resistance proportional to the velocity, you may derive analytical expressions for the curve, that typically(if I remember correctly) involves the hyperbolic functions and their inverses.
Unfortunately, for the vast majority of cases, this is not a physically realistic solution, since air resistance is proportional to the square of the velocity (and there is no nice, analytical solution for that).
My point being that if you care about physical reality, rather than mathematical prettiness, the existence of a solution for drag proportional to velocity isn't terribly useful or relevant.
Some objects (mainly small objects) can have laminar flow where drag grows linear with the velocity. Unfortunately, their typical timescale is so short that you cannot really call their motion "falling".
This question is not well posed. When there is air resistance, there is also lift. Both depend on the shape and the mass distribution of the body, and its orientation and rotation. These effects can be very significant, in which you can certainly convince yourself by comparing the flight of a glider and a ball.
It's not so much that the flow is laminar, it's that the flow (for very low reynolds numbers) is dominated by viscous, rather than inertial forces. Laminar (but inertially-dominated) flow still has drag that scales as v^{2}. As you noted, in air, the only time you would have viscous-dominated flow is for very small, slow moving objects. There are however some physical cases where this could occur at larger scales. In a fairly viscous fluid (corn syrup, for example), a BB, marble, or even golf-ball sized object could have drag that scales with v rather than v^{2}. Most of the time, though, when people ask about drag, they aren't thinking of a marble falling through corn syrup
Quite so. And? It doesn't follow from this that what I wrote in my post was wrong. I pointed out that there were cases with air resistance in which analytical expressions could be found, I did not say that that was generally true.