You need to be a Supervisor First.
See Related Link on Becoming a Supervisor.
Yes, once you are a Supervisor, you can request to supervise any category you wish.
Tricky one to answer. Supervisors don't supervise a set number of contributors - they supervise individual categories. For example - I supervise the Snakes category. Some categories will be more popular than others - and thus attract more contributors.
You can go to browse categories and click on which one you would like to supervise, then click "supervise this category" which will appear on the right side of the page.
The best way you can request additional categories to supervise is by contacting Ganderton, via his message board.
If a category isn't supervised, then there isn't a Supervisor who is willing/able to supervise it.
Sadly, no, you cannot. Another Supervisor might want to supervise that same category, so no 'rain checks' are allowed for categories. However, when you are old enough, you can co-supervise a category with another category supervisor for that same category you wanted to supervise.
A supervisor can take as many categories under his or her wing according to how capable the supervisor feels about handling them. The sky's the limit.(See the Related question.)
Generally just try and keep the category(s) as clean as possible. Generally you focus your time on these categories.
Banning users, removing questions and answers, find vandals, supervise categories and merging and splitting answers
No, not at all! The category topics you supervise can all be totally different topics. Just choose whatever you like.
Well, yes, they can watch over categories and answer questions in them, watch for vandalism, etc. but they choose not to mainly focus on a few categories in specific, unlike Category Supervisors. Rather they choose to roam around, hence the name.(See the Related question.)
Procurare is supervise in latin