There are no problems under Windows 7 64 bit, can you please describe, precisely, what's happening after you select "Register Serial" ?
If FSX returns back to flying without doing anytying, it means you entered your Serial wrong. If you could contact me via email, I can check your order and your serial.
If you entered your serial correctly, you should get a request "Do you want to activate on this PC?", and "Are you sure?"
If you reply YES, and THEN nothing happens, it's something related to the internet connection, like firewall, etc.