A paper wallet is basically a money grab. It's used to sell stationery and holographic stickers like you see on that site you linked to above. The thing with paper wallets is that they are very limited in two main ways a) you get only one address per paper wallet b) spending from them is a bitch. Regarding b, if you don't spend from them correctly you can loose your money.
If you have only a small amount of bitcoins then storing them on a desktop wallet is fine. For larger amounts look into offline Electrum wallets.
Sort of true. But it can be useful to have a paper wallet backup like the one Blockchain.info provides (as long as you keep the paper wallet someplace safe).