Since rooting hormone helps cuttings root, would it help plants build more/stronger roots if you watered with it?

That depends on the rooting hormone you use. The bottled kind are generally too powerful to use that way, and I wouldn't suggest it at all. You can kill plants by overdosing the rooting hormones. However, if you are using willow water as a rooting hormone, then, yes, it does help plants to get off to a better start...but so does using rock phosphate right in the root area.

Not really. Rooting hormones are those thingys that help plants develop roots where none exist and are present on all plants in certain areas of the plant (where roots will develop in the right conditions) and putting more of those hormones in those areas aid root development, initially. Once the roots do develop the auxins do little to promote root growth.

I put a couple teaspoons of rooting hormone to a gallon of water, per directions on the bottle, and water seeds and cuttings with it. I can't remember what brand my DH picked up but I think it helps a lot.