Will addresses show up only after it is done syncing?
There have been lots of complaints of people that tried to receive bitcoins but only realized then that they wouldn't be able to spend them once Bitcoin Core is synchronized. This is why there is a push to hide receiving addresses until synchronization is complete or at least better explain what's happening. I thought that those changes would only be added with the 0.13.0 version of Bitcoin Core, which is not out yet, so you should be able to create a new address in the "Receive" tab.
Be aware that you'll see incoming payments and will be able to spend your balance only after your Bitcoin Core has caught up with the blockchain to the point where the payment was confirmed on the blockchain.
If you want to immediately start spending your bitcoins, you might want to look at a light wallet such as MultiBit HD or Electrum instead.
a bit off-topic: Is it safe to just copy/paste the wallet.dat around or is it any risk if the sync is at a certain state while I copy/paste the wallet to/from the bitcoin data folder?
You should only move/copy/paste your
wallet.dat while Bitcoin Core is not running.