AirPort - "Remember This Network"
As for the first problem, look at the attachment. If your computer is not set to remember your network, tell it to remember it, and it will remember the password, as well. That is my experience, at least.
The second problem should be fixed by fixing the first. Every so often, the AirPort on your Mac will rescan to find more networks. Judging by your problem, it asks for your network password every time it does this. My MBP must disconnect and instantly reconnect, because I don't notice it when it scans for networks. Yours probably tries to connect again, but it doesn't remember the password.
One question, if the above doesn't work - do these problems only occur when you are on your network, or on all networks?