Is it a good idea to keep a paper record of my Bitcoin wallet address and the passphrase?
That depends on how likely it is that someone untrustworthy accesses both the written passphrase and the wallet.dat file.
If I add bitcoin to the wallet and backup the wallet to say a USB stick and my computer gets hacked and the wallet stolen, will I have access to my bitcoins via the backup and / or paper record
Yes. If you have a recent enough backup and you know the passphrase (or can find it on the paper) then you will have access to any bitcoins that have not yet been taken from the wallet.
or will whoever hacked my computer have control of the wallet?
If they can guess your passphrase, or if they capture your passphrase while you type it in, or if they can get access to your written passphrase, then they will have control of the wallet as well. If they take the bitcoins out before you can access them, then you will no longer have access.
And is there anything else I should do before starting to trade to secure my Bitcoins?
That depends on what your plans are.
One other thing. If someone sends a bitcoin to my wallet address how will I know when it has arrived? Will it just appear in my wallet
Yes, as long as your wallet is communicating with the bitcoin network at the time the transaction, it should just appear in your wallet. If your wallet is not communicating, then the transaction will show up once it is confirmed and the wallet has caught up with synchronization.
You'll want to make sure that you understand the concept of a transaction being "confirmed" and the importance of confirmations when transacting with someone that you don't already have a trust relationship with.
or do I need to check a central registry?
Bitcoin is decentralized. There is no official central registry. The closest thing to that concept is the blockchain that every full node peer maintains its own copy of.
