The bitcoin price is you have to agree on and it will be fixed for a certain amount of time, just enough for you two to be able to complete the transactions.
If you agree on 4.5 btc then thats what you're supposed to get period.
Bitcoin will be sent to whatever address you provide, so if you provide a coinbase address you will get btc to that address, if you provide an address generated by multibit/electrum you
should get btc in your multibit/electrum wallet.
While coinbase is a pretty popular option there are many bitcoiners that don't use them and moving funds in bitcoin is possible without them.
I would recommend you get a wallet like multibit/electrum and play around with it till you get the ropes. Once you receive your btc you can exchange them trough the exchange of your choice
Here you will find a list of wallets you can chose from:
https://bitcoin.org/en/choose-your-walletHere you will find a list of exchanges to get you started:
https://en.bitcoin.it/wiki/Trading_bitcoins