Contact between matter and anti-matter causes a "matter anti-matter annihilation" where both are momentarily converted into energy and then into other particles. They don't just cancel out. If one atom of anti-matter were to fall out of a "penning trap" and were to get near enough to an atom, it would only interact with one atom (well also depending on the mass of each). It would not destroy the world or anything.
I don't know what you mean by "connected" but I don't see how that logically follows any of your premises. If by connected you mean, all in the same universe, or all came from the same place, then that may be true for other reasons.
When an atom of an element undergoes radio-active decay, it doesn't just disappear. It ejects a neutron (depending on the type of radio-active decay actually) and becomes another element. Because it's loosing a neutron, the decayed product is lighter. Also, a half-life doesn't expire. A half-life simply means that if you take a chunk of something radio active, how long will it take for half of its atoms to decay. After the first half life, you are left with 50%. After the second half life, you are left with 25%, etc.
Despite being taught that there are three states of matter in school (at least back in the day), IIRC there are 5 (am I forgetting one?). You have the condensate, gas, liquid, solid, and plasma.
If you want to know more about particle physics, I highly recommend:
http://www.particleadventure.org/