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you can get touch up paint from one of the corvette mail order catalogs or any Chevy dealer. Remote battery at any drug store or home center (be careful opening and removing ang reinstalling. As for flat tire messge do a search on sensors or low/flat tire message on DIC. It could be a number of things you may have no sensor, bad sensor or possibly sensor needs to be retrained. Good Luck enjoy.........
To answer "3. On the remote, what does the "passive, on/off" mean?"
Early C5s had a remote which would unlock and lock you car as you approached and left your car with your fob in your pocket. If you locked your keys in the car, you could just shake the car and door would unlock (a motion sensor in fob sent out signals to lock/unlock).
BTW, you may not need a battery for your fob if it isn't working. Try holding down the lock and unlock buttons at the same time for about 10 seconds or until the horn blows, if horn blows you have re-synched your fob. You may have to relearn your fob also using your DIC.