So I've downloaded and run the Bitcoin Core client and synced it, set up a passphrase and backed up my empty wallet and written down my privatekey.
Do I need to anything more before buying bitcoins?

I've looked at Localbitcoins for buying bitcoins - but I know they have a fee. As soon as I make a user at their site I assume I get a new wallet and new privatekey through them? And if so, can I then transfer those bitcoins to the wallet I have in Bitcoin Core without having to pay a fee over again? after all those are both my own wallets.

And last - I don't have an offline computer - but want to keep my wallet stored and backed up safely. As I dont intend to use it all the time but store it I'm thinking if I can simply compress the wallet.dat with a strong password, and store it at two locations thus meaning I both have a backup of it and also keep it away from access to others - will this work? There is no need for the wallet or so to be connected to the grid? If I later 5 years from now uncompress the wallet.dat onto a brand new computer I will still be able to access the bitcoins in the wallet provided I have both my private-key and my passphrase ? Do I need a second thype of key or address?

Hope those are questions easily understood - please reply if you know the answer, or ask away if there are some of my questions that are hard to understand