What is a bitcoin wallet? I do not understand how addresses and wallets are related. I see that you are advised to generate an address for every transaction and, furthermore, use (multi-in multi-out) tumblers to maintain anonimity. This will create a hell amount of (unspent) keys+addresses. How do you manage them all and handle complexity? How do you remember that this was a simple transaction from that wallet to mine and the rest is just intermediate activity. Is a wallet just a collection of (unspent) addresses so that you can simply merge different (wallets of) bitcoins by just gathering these keys/addresses? I understand that to have a coin means that you know which address has some unspent bitcoins and you have a key to demonstrate that it is your coin/address in the next transaction. But what if you share same coin addresses between wallets and one wallet will think that there are still money while you spent them (peahaps partially)? What about compatibility of various wallets, is there a strandard/converter? Am I right that the thing that you must keep in mind is the list of addresses plus corresponding keys and the wallet is such a record?
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.