What happens when someone sends bitcoin to my wallet? I didn't get it. Should my wallet be online? What if it becomes corrupted before I make a backup? Can I use my backup wallet that was backed up few days ago?
It's actually quite simple. Your wallet is a collection of private keys, each identified by an ID. When someone sends you bitcoins, those are just assigned to one of those private keys. 

The information about which key has how many bitcoins is stored in the bitcoin network itself, ie in the database each client has.

Now if you want to use the bitcoins which belong to one specific key, you need this key to sign the transaction.

So no, you don't need to be online to receive a transaction. You can also restore an older backup, as long as you haven't created any new keys in the meantime. If you have created new keys, it all depends on your luck - was the transaction sent to a new or an old key?

And if it gets corrupted before you make a backup you are SOL - nobody has the correct private key then.