Softboxes can be put over continuous lighting or strobes. The strobe will be way way brighter than regular light unless you have super weak strobes or your regular light is ridiculously bright.
Yes the flash goes in the softbox. If you have a modeling light on the strobe, it will turn off while the strobe fires. If it did happen to stay on, the strobe would be so bright it would render it invisible to the camera, like peeing in a pool and expecting to see it. For photography, softboxes are used to diffuse strobes. Continuous lighting with a softbox is more of a video application. As far as the 2nd question, I'm not even sure what you're asking. If you turn down the power on the strobe & or modeling light, it will put out less light as one would expect. The light from a softbox is from a strobe (or other flashing unit) almost all of the time in still photography.
