This is my second 2002 VW Cabrio, and when the problem appeared on the first car, I was far more concerned. The Check Engine code P1582 is apparently (somewhat)common with the model, and is related to a "fuel pressure" sensor that has gone bad. That COULD indicate a fuel leak, but more than likely just involves replacing the sensor if everything tests out okay.
Not that I recommend it for everyone, but I bought a OBDC-II reader rather than dealing with this issue immediately. That way, I know if OTHER diagnostic codes appear, and I can clear the annoying false-positive indicator. Your mileage may vary!
My check engine light has been turning on and off for four years now. I have no idea why except for the fact that it's on more often in hot weather, and off more often in cooler weather. There are no other driveability issues, however.
3 years ago I have had the same problem with the oil light & beeping noise and it turned out to be my transmission fluid was low. Lately after 3 years it happened again & forgetting I took it to the shop & they claimed it was an oil leak. They claimed to have changed a whole mess of things $500.00 worth of problems yet the light & beeping noise still sporadically came on. I finally remembered that 3yrs previously I only added transmission fluid so I tried that again to find, that there is a leak in the transmission. Hopefully It won't be too costly & this will solve the issue.