The QR code is simply a convenient way of sharing an address, to avoid having to type it. It encodes exactly the same information as the alphanumeric address and provides no additional security. If you have another way of giving someone an address (e.g. sending it to them by email), there is no need to use a QR code at all.
How the other person verifies that the address is yours is something that you have to solve for yourself; Bitcoin itself doesn't provide features to help with this. For instance, you could send the address via a PGP signed email, or make it available on an HTTPS website, or ask the person to telephone you and verify the address. But it is an important issue; if an attacker impersonates you and the other person pays their address instead of yours, the attacker will get the coins and there will be nothing anyone can do about it.
There is nothing you need to do after receiving coins.