If you enter the status window, your pet cannot die. This means that a couple of cheats are available. 1) If your pet dies, restarting the game and quickly entering the window before it dies again means that, if you exit the window a bit later, your pet is healthy and ready to carry on as normal, or 2) You can have a tamagotchi live forever. You need a Game Boy adaptor for this to work. Go to the status window, and leave your game boy plugged into the wall for as long as you want.
Hatch a pet from a black egg, and feed it nothing but carrots. When it is 6, it will turn into Mimitchi.