I've been doing a bunch of rack research lately, I would suggest you spend some time on the Thule and Yakima websites as most of your questions can be answered with research at those sites.
Questions: Can I use the factory rails with self-installed cross bars as the foundation for my carrier?
A: You have the option of using the Ford cross bars or a third party cross bars. Advantages of using the Ford cross bars are that they will look better when you aren't using the rack, as well as being more aerodynamic (less wind noise) the advantage to using 3rd party crossbars is you will have more cross bar area to work with. Jump onto the Thule website and use their configuration tool for the Edge with the factory rails and it will show you the options I've just outlined.
Q: Anyone recommend box vs. basket/bag configuration? Manufacturers? Styles and sizes?
A:Thule and Yakima have boxes that will attach to the Ford crossbars and bags that will attached to the Ford rails. I've done some measurements I can share with you. If you open the hatch, and then measure the maximum length box that can fit between the raised hatch and the antenna in front you'll find it to be approximately 58". You also might want to consider adding a Stubby Antenna to maximize the length of the box you can use. With a stubby antenna installed it looks like you can max out with a box approximately 75-78" depending on how much overhang of the windshield you are comfortable with. Remember, Ford rates the maximum amount of weight to put on the roof at 100lbs so pack your box using that in mind.
Q: I may someday want to put kayaks or bikes up there and believe Thule or Yakima probably has systems that clamp to the elliptical cross bars? Am I correct?
A: Yes, you'll find that bike, kayak and ski attachments from Thule and Yakima are available that will attach to OEM elliptical cross bars.