Rookie discussion here. For the sake of this discussion, let's define a revolution as the a couple 360 turn of the gear/sprocket and a firing cycle as a complete 1-4 cylinder fire. Okay, so on my 1996 Camry, the cam gear on the timing belt turns a complete revolution 1 time for every 2 times the crank sprocket turns. So, does that mean for every 360 revolution of the crank sprocket I get 1/2 a firing cycle (2 revolutions=1 cycle), or for every revolution of the cam gear I get 2 firing cycles?

So, does that mean for every 360 revolution of the crank sprocket I get 1/2 a firing cycle (2 revolutions=1 cycle), or for every revolution of the cam gear I get 2 firing cycles?

"...for every 360 revolution of the crank sprocket I get 1/2 a firing cycle (2 revolutions=1 cycle)..."

4-Stroke engine has 2 strokes per crank revolution. Two revolutions to get back to the beginning of a cycle.

http://auto.howstuffworks.com/engine1.htm

In contrast, 2-Stroke engines fire once every revolution of the crank.

http://science.howstuffworks.com/two-stroke2.htm

So, from a timing standpoint, if I brought the cam and crank to TDC, pulled the belt, turn the crank 1 revolution, and reinstalled the timing belt, would that throw the timing sequence off, or would it stay timed correctly because the crank is simply moving the pistons up and down?

It would stay in whatever timing was originally set. There is no such thing as TDC for cams, only pistons.

Timing would not be affected. The crankshaft is in the same position (zero degrees TDC) at both 0 degrees and 180 degrees of camshaft angle.