So if you would buy some tokens right now, do they stay on that website or can we download a wallet somewhere and send it there immediately?
It's the exact opposite of what you think. The tokens
do not live on the website ever. The tokens live on the Ethereum blockchain and you can access them using any ethereum wallet. We've tested using Mist wallet, MyEtherWallet (and Saturn is natively supported there), MetaMask wallet (you can store tokens there but cannot send them) and imToken mobile wallet.
Will give a development update in a separate post.