Let me take a shot too. I'll probably just confuse you more

1) You can measure the speed of light directly. I did it in a second year university course, 35 years ago, so I bet they do it in high school now. What we did was send a beam of light through a slit in a piece of paper (I bet they use lasers now), then to a mirror that was mouinted on a motor so that we could make it rotate on it's vertical axis (but it was kept stationary to set up the apparatus). We arranged it so that the beam reflected off that mirror to another mirror a long (but measured) ways away, which reflected it back to the rotating mirror, and then back to the source, where it showed up projected on the piece of paper that the slit was in. Two mirrors and a motor, simple. When you turn the motor on so that the first mirror rotates really fast, the projected light on the paper starts to flicker, because most of the time the mirror isn't lined up right any more. And the flickering light moves off to one side, because, when it is lined up right, in the time that the light takes to go from the rotating mirror to the distant mirror and back, the mirror has turned a little bit, so it doesn't reflect it right back where it came from. You can measure the amount it moves to the side, and the rest is just geometry. I think we got the answer within 25% or so that way. Not the greatest accuracy, but they have much better ways with more sophisticated equipment too.

2) Why do moving objects gain mass? This one is really easy. No, really.

Energy and mass are equivalent. That's what e=mc^2 means. When you add energy to something you increase its mass. It's obvious in nuclear energy, where we take energy out of Uranium, and decrease its mass. And its also true of moving objects, when we put kinetic energy into them, we increase their mass. By exactly the amount that equation says.

So the question is really self explanatory. Motion is energy, and energy IS mass. they are the same thing.

3) Nothing magic about 1/3 c. Objects gain mass as soon as they start to move. But that equation (e=mc^2) means that they gain very little mass until they reach pretty high speeds. At the velocities we experience in every day life we can ignore the effect. At 1/3 c (or so) the effect becomes big enough to make a significant diffrerence.

4) Relativistic velocities don't just add arithmetically. They add using something called the Lorentz transformations. If you can stand the math, this page gives a reasonable explanation. If you can't stand the math, special relativity will probably remain a mystery. It really is counter intuitive.