The answer to your first question is a little tricky, because not everybody uses the word weight the same way. There is a force of gravity on the falling object, so in that sense it does have weight. However, gravity isnít actually a force you feel. You feel a floor or a chair pushing up on you. So if by weight you mean a force which keeps things from falling, the sort of force you can feel, the falling object is weightless.In practice, if youíre on an airplane in free-fall, you do feel weightless, just the same as you would if there were no sources of gravity around.
Gravity is so unusual this way, being something that you canít feel, that it turns out in some ways best not to think of it as a force at all. The reason that an object will take the path it does in a gravitational field when no forces are on it is that space and time are bent so that the natural "straight" paths are those that freely falling objects take. This is one of the main ideas behind Einsteinís general theory of relativity.
For most practical purposes, however, you may use Newtonian mechanics with a force of gravity giving rise to the familiar acceleration dropped objects undergo. This force will be less at high altitudes than it is at low altitudes.
For your second question, we can appeal to the principle that the same thing must happen to the projectile (for example, it must land in the same place), no matter who is looking at it or how fast the observer who is looking at it is going. The other half of this is that the laws of physics in this moving frame of reference are the same as those in any other frame of reference (F=ma, still, for example).
We can imagine an observer who is moving horizontally at the same speed the projectile is initially moving in the horizontal direction. Then the moving observer will just see the projectile being shot straight up. If there are no forces in the horizontal direction, the projectile should go up and come straight back down with no horizontal deviation, just as if the observer and projectile were at rest when the projectile is fired. We can then translate all of our coordinates so that the projectile has a component of motion which is horizontal and a component which is vertical, and the result is that we get the same answer as if the horizontal and vertical components can be solved independently.
With more math, Newtonís law F=ma is whatís called a "vector equation", which has three components: F_x=m(a_x), F_y=m(a_y), and F_z=m(a_z). This is the simplest way to describe what we observe in nature.
Real physical situations however can couple horizontal motion with vertical motion. Introducing air resistance to the projectile problem, for example, will mean that the horizontal and vertical components will no longer be easily separable. But one can always break the force down into its three components and relate it to the three components of acceleration, even if it changes with time.
Tom J. (w mike)
(published on 10/22/2007)