I have read that pouring a mixture of 3T hydrogen peroxide in 1 gal water onto the soil of my plants will help them grow because HP has an extra oxygen atom that will help the roots. Any truth to this? I may have the HP : H2O ratio wrong.

There is truth in there. Hydrogen Peroxide (H2O2) is water with an extra oxygen molecule. Its pretty unstable and will release that extra molecule when mixed with water. So there will be a little bit more oxygen at the root zone which may help prevent root rot and encourage root growth.

I have used a drop of hydrogen peroxide in water for rooting cuttings and it works really well. I have not used it for watering my already rooted plants. I would think that the only advantage to watering soil with H2O2 water is if you overwater and want some protection. The cause of root rot is a lack of oxygen at root level. That is caused by overwatering. Oxygen at root level will encourage root growth but so will not overwatering in the first place.

There are a lot of different strengths of hydrogen peroxide for sale so a blanket statement of 3T/gallon could be way too much.