Hydrogen Peroxide (h2o2) isn't bleach (NaClO), and in fact they are as dissimilar as is bleach and water. You can even drink a considerable amount of 3% h2o2 (over the counter typical strength) without any adverse effects (though I don't recommend it, some even do it for health reasons) ... bleach on the other hand is just bad to consume at any concentration and putting bleach on your strawberries would be a bad idea.
Hydrogen peroxide won't affect the taste of your strawberries and there is no need to rinse them off. I actually soak our strawberries with ascorbic acid (vitamin c) after picking for about 5 minutes and they last many weeks without mold. Without it, mold usually sets in within 1-2 days. While ascorbic acid is good for you, if you soak your strawberries too long they will get a bit of that Vitamin C taste, which I don't care for, but a quick soak works fine, and I follow it with a very quick rinse.