No. Baking soda will not hurt your gums. Instead, it can increase the pH of your mouth which in return, decreases your risk of cavities if you do everything else right.
Hydrogen peroxide can help whiten your teeth but it will take a long time considering that the concentration of the over the counter form is only 3 percent. The other problem I see is that long term use may kill the good bacteria in your mouth leaving you vulnerable for opportunistic infections. Oxygen can be, believe it or not a poison for live cells!