A few seconds after the sender sends them.

Because they appear in his/her wallet.

Technically: because his/her bitcoin client (wallet software) notices the transaction that was sent throughout the network. Note that you don't have to be online in order to receive bitcoins. The sum of bitcoins is added to your address, regardless if you're right there and then to notice, or don't connect to the network until a week later.

Same as what will happen to the Euro and US Dollar if there are no banks, ATM machines, credit cards, paypal, and cash registers anymore. Extremely unlikely, but it would die.