There's nothing magic or mysterious about relativity. Among other things, it's based on the idea that fundamentally you've got to worry about the issues that arise when you try to quantify time and space.
It starts with the assumption that anybody's idea of time is as good as anybody else's, provided they're not accelerating. So if I've got a really precise watch, and I measure a time interval of ten seconds between two events, then those two events really are ten seconds apart. However, I have no guarantee that other observers will make the same measurements.
The next basic assumption is that light travels the same speed in a vacuum, no matter what. That is, no matter how fast you (the viewer) are moving, and no matter how fast the light source is moving. The view is supported by the theory of electromagnetism.
Before I go on I'd like to point out that you're trying to show that relativity is internally inconsistent -- that is, that it says two things which can't both be true. In other words, you aren't claiming that relativity disagrees with the way the universe really is.
Getting back, this is where your idea breaks down. You're talking about what happens at the speed of light. But if I get in a rocket which then flies away from the Earth at the speed of light, then according to relativity, I am not moving at all -- I am perfectly still, and the Earth is flying away from me at the speed of light. Since I'm perfectly still, I don't have to worry about time ceasing to exist or anything like that.
Along the same lines, if the travelling twin became one second older, what that means is that a clock that he carried with him ticked one second. However, the clocks that sayed behind on Earth would have ticked much more than that, so he could have moved quite a good distance away in two years (much more than 300,000 km).
Anyway, you need to correct your understanding of relativity if you want to really set up a solid case against it.