What does the doorbell sound mean while in the gardens in Sonic 2 Battle..

If I remember correctly, it means one of your chao died. It happens like this: your chao is upset for whatever reason. It sprouts a pink and white transformation bubble. After a minute, if you hear a doorbell sound, check on the transforming chao. If my memory serves, once the bubble is gone, the chao will also have disappeared without a trace! Be sure to check about that! No chao fan wants their favorite chao to disappear into the Chao Sky Garden. (Or Chao Ground Garden if a Dark chao dies.)

When your chao is in a cocoon and hear the bell that means a chao of yours is dying. Or if you hero garden, you might hear that bell sound.

It means the chao is (either) reincarnating or dying completely. I have never let the latter happen to my chao, sticking close to an entrance so I can go through it if I see a grey cocoon. The pink cocoon is the chao reincarnating, and leaves an egg behind (irrespective of the two parent chao, it looks like a normal egg, rather than the coloured shiny chao eggs that sometimes appear). It is at least one of the two, although neither has happened for a while so I cannot remember whether or not it is both. Just look for a grey or pink cocoon when you hear it (use the camera swivel)