Regarding launch pads, Florida was chosen for two reasons. First, launching over the ocean is much better than launching over the continental U.S. if there is a crash. You don't want debris raining down on Denver, Kansas City, St. Louis, or Chicago. (Well, maybe Chicago would be OK, no one would notice with all of the shootings going on.)
The second reason is that the earth is spinning faster the further south that you are (closer to the equator). Therefore, the further south that the launch pad is the more of a "boost" one gets to achieve orbital velocity. https://www.livescience.com/32721-why-are-rockets-launched-from-florida.html#:~:text=It%20was%20selected%20for%20two,west%2Dto%2Deast%20spin.
I'm not sure what an "on-site" satellite is? If you could clarify that term, please do. The fact is, communications technology has reached a point that you can ping a satellite from anywhere using relays. Furthermore, we're really only talking about the SpaceCommand HQ, which may not be an operational base, just a place for the brass to hang out with their wang out. Finally, you wouldn't want just one base that can talk to your satellites, you need back ups and back ups to your back ups. Therefore, like other branches of the military, they'd have operational bases all over the world.