Whether you are a Soul Reaper or not is decided at the graduation exam. There are exceptions to this, like if someone shows extreme promise beforehand. No, you have to be able to use your shikai to become a shinigami, so you get your zanputo at a certain point in your training. How you obtain it remains a mystery. I believe its a physical manifestation of your spiritual energy, but I may be wrong.

No, there was an episode where they explain Rukia's mission. It was episode 109. However, I can't quite remember it at this point.

Ah, the fight beween Ichigo and Kenpachi is breathtaking. You'll love it for sure!