1)

No.

Receiving addresses are not shown because they are intended to be used only for a single payment. The practice of reusing addresses weakens your privacy, but also that of others in the system who you interact with.

It is possible to see the receive addresses that were created in the past, but you need to explicitly check the address reuse box in the receive payment tab.

2)

You can move the wallet.dat file when Bitcoin Core is not running. You can also copy to make a backup, but realize that the backup will need to be recreated every 100 incoming payments (this will change in 0.13, where a backup will be permanently valid). Do not actually use copies of a wallet from multiple locations at once.

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.