There isn't any way to send or receive Bitcoins without using addresses, but if you use BIP70 Payment Requests, then you won't need to give anyone an address. The address will be part of the payment request and most modern clients will do all the work for you. You just need to allow the send.
I think we should make this type of platform like i described above!
I am interested in developing it
Well I think the payment requests can already do what you describe.
You create a payment request and upload the request file to your site. If someone wants to pay, then they download that request and their client immediately recognizes it as a payment request. It will know where to send the Bitcoin and how much to send, so all the user does is enter their wallet password and hit ok. Then the client just sends the Bitcoin normally.