Having some issues with our sonata

My moms sonata more specifically, she complains of 2 things. One being that the gas gauge seems to be going wild, she has stated that a couple times it just turned off and dropped all the way down to E, right after she put 15 bucks worth of gas in the car, and i myself have seen the gauge wildly fluxuate between E and Full, far too much to be just the bounce of the float in the tank. Im not sure if the thing in the tank itself is bad or if this is an electrical problem. The other problem she is complaining about is that twice when she got out of the car, after taking the key out, the car made a ding ding ding and would not allow her to lock the car doors or anything. My guess is the ignition switch, is staying stuck in the on position after the key is removed but i figured id get some feedback first. Any thoughts? How do you think i should attack this.

The gas guage problem points to an electrical problem as you guessed right... I have a pickup truck ( ford ) that did the same thing also .. needle would go all to full and then down to 1/4 tank after about 5 minutes.. then jump back up to full again or sometimes to just a half tank even though I always keep the tanks full ....... The problem I found was that a wire behind the dash panel was touching metal lightly and had rubbed the coating off the wire and it was causing the guage to breifly short out ..... just put some electicial tape around it and made sure it was free from touching anything.. fixed ........... of course thats one thing to look for anyways , plus make sure the wires in back of guage are firmly connected to it also ......... I doubt also that the problem is the float in tank ... but always a possibility of course..

Even if the igintion switch is 'stuck' in the on position I would think you would still be able to lock the doors ... There has only been one time I was not able to lock my doors after getting out of the car and the cause was that a back door was not all the way closed and my anti-theif would not engage until I had it closed right ... I also kept getting the ding ding ding warning too