What is a bitcoin wallet?
You are asking a lot of questions in one post but to answer the question in your title concisely:

A Bitcoin wallet is loosely the equivalent of a physical wallet on the
Bitcoin network. The wallet actually contains your private key(s)
which allow you to spend the bitcoins allocated to it in the block
chain. Each Bitcoin wallet can show you the total balance of all
bitcoins it controls and lets you pay a specific amount to a specific
person, just like a real wallet. This is different to credit cards
where you are charged by the merchant.

From Bitcoin.org.
A wallet contains any number of public/private keys.  You can easily create new wallets or new public/private keys in your existing wallet.
I'd recommend breaking down anything still unanswered into their own specific question.