So, that Umbrella bracket will hold both my umbrella & flash? Is the 6' height max on the stand a good enough height (better than my tripod, I know, but if i'm going to get a whole new light stand I'd rather get one that will work with taller people too)?
Yes, that bracket will hold your flash and your umbrella. If you look at the photo of it, you'll see a little egg shaped hole. That is where your umbrella shaft goes thru.
The six foot heigth would likely work for most of what you would be doing, but obviously there is a reason that 10' and 13' are popular heigths too. One thing about all this is that you will be buying more stuff as time goes by if you expand what all you are shooting. So, the shorter stand will come in handy for one thing or another later on. For 20 bucks it's pretty hard to go very far wrong.
You approach is still sounding good. Start with your one umbrella and a reflector, and see what you can do, then go from there.