Yes, exactly. However, Albert Einstein beat you to this discovery by about 100 years with the equivalence principle. The key idea is the equivalence between a downward gravitational acceleration and downward force due to an acceleration upward. There is no experiment you can locally perform that will tell you whether you feel heavier because the elevator is accelerating upward or because the gravitational field below you just got stronger. Since weight is just the force due to gravity, then you can use Newton's second law to calculate your weight in the elevator as $F = m(a+g)$, where $g$ is the acceleration due to gravity on the surface of the earth, 9.81 m/s^2.
So yes, everything in the elevator feel heavier when it's accelerating upward. When it's accelerating downward, everything gets lighter (unless of course it's accelerating you faster than gravity would, in which case you'll feel pulled to the ceiling. You'd need a cable pulling the elevator down for this to happen.). In fact, if the acceleration downward perfectly matches the acceleration due to gravity, you'll be weightless. This is the second piece of the equivalence principle - an elevator accelerating downward at the same rate as gravity is indistinguishable from floating in empty space with no gravitational field at all.
For some more about the equivalence principle, see here:
The reason jumping won't work is because you'd need to time it perfectly - the chances of you doing this are impossibly small. I'm not 100% sure, but I've always heard the thing to due in a falling elevator is to lie flat on the floor as to distribute your weight to spread the force of impact out across your entire body.