This is probably the hardest part about General Relativity (GR) for people to understand. It's taken me many years and a lot of learning to finally come up with an interpretation that makes sense to me. The most insightful aspect of GR is actually the mathematical description. It's pretty abstract at first and takes a lot of mathematical background to start learning it, but once you have a pretty good grip on it, the math actually reveals some great insights.

First of all, when Einstein developed Special Relativity (relativity that only applies to non-accelerating systems), he formulated it in four dimensional spacetime, where time was treated just like another dimension of space. The only difference was that time is a dimension that you can only travel in one direction: forward. To me, this makes time really simple: everyone and everything is travelling through time, just as if you were traveling through space, only you can't turn around. This, combined with the fact that light travels at a constant speed, is the basic premise for Special Relativity. It also comes with some interesting results. As I said before, everything is traveling through time, but at what speed? It turns out, if you say that something sitting still is traveling through time at the speed of light, then everything works out nicely. This is why nothing can travel faster than the speed of light. Essentially, when you begin traveling through space, you are actually changing direction, thus you are not traveling through time as fast (time dialation). So you are always traveling at the speed of light, but some of your speed is in the direction of time and some of it is in one of the three space directions. If you are traveling through space at the speed of light, you are no longer traveling through time, and since you can't go back in time, that's the fastest you can go. So from this perspective, the speed of light is not the fastest speed possible, it's the only speed possible.

Now that's only one aspect of Special Relativity. Things also change depending on you frame of reference and so on, but lets continue and see if I can answer you questions about gravity.

Einstein wanted to extend his theory to things that are accelerating, so he developed General Relativity. Einstein found that things in the presence of mass are affected just as if they were accelerating (equivalence principle). From this he saw that there was a relationship between mass, energy, and the way stuff moved. He was then able to clump all this stuff together into one big quantity called the Stress-Energy-Momentum Tensor. I sort of think of this as a mathematical description of the physical state that something is in. This single mathematical entity describes all the imporatnt stuff about a particular collection of energy - how much energy is there, how is it distributed, how is it moving, pressure, etc. Also, since energy is equvalent to mass (E=mc^2), it also tells you how much mass is there.

So, by extending his relationship between movement and spacetime to energy and spacetime, Einstein was able to show mathematically how energy influenced spacetime, or rather how energy moves through spacetime.

To answer your questions, time is not a side effect of warping space, it's simply a direction that energy travels in. If you remove all the mass (and energy) in the universe, then there is nothing to move through time, so time becomes irrelevant. And finally, yes, time, space and gravity are all affected simultaneously by the amount of energy present and the "state" that it is in.

I hope this helps.