I've been using the setup you're considering now and love it. I use the MLT described, a propane burner, 20qt stockpot, and keggle. If you can avoid it, don't carry hot water through the house, it's just asking for an accident. What I do is use my stockpot on the propane burner for heating all my mash/sparge water, and drain the MLT into the keggle. To save time, once my last sparge water has been added to the MLT, I throw the keggle on the heat to start it on its way to a boil.
For your hose water question, you have a couple options. The first and hottest water usually goes into my MLT (after the grain is emptied, of course), which gets some oxyclean added to it, and all my brew-day equipment that fits gets thrown in there for a good cleaning soak. Since you can't let the water just run off into the carport, maybe you can try a pond pump and a cooler or bucket full of ice. Since the ground water temp here in Florida tends to be too hot most of the year for adequate beer cooling, I use one of these
submersed in a cooler full of ice and some water to pump chilled water into the wort chiller, with the output water recirculated back into the cooler. I usually don't switch to that setup until the wort is somewhat chilled, so as to not waste ice, but as long as you get enough ice, it should work fine for you. Best of luck man!