The key fob issue could be low battery on the fob itself.
I haven't had any issues that resolved itself in the manner you described. Usually, I've needed to shut down the car, wait a few minutes for the system to completely shut down, then restart.
I believe software updates are "over the air" (OTA). I don't remember all the exact menu names, but if you go into settings, there is a menu item that says something like software information. Under that option you have the ability to "check for updates". I believe your car must be connected to WiFi for this to work (either the one that comes with OnStar, or you can also connect the car to your own WiFi at home).
I haven't had any issues that resolved itself in the manner you described. Usually, I've needed to shut down the car, wait a few minutes for the system to completely shut down, then restart.
I believe software updates are "over the air" (OTA). I don't remember all the exact menu names, but if you go into settings, there is a menu item that says something like software information. Under that option you have the ability to "check for updates". I believe your car must be connected to WiFi for this to work (either the one that comes with OnStar, or you can also connect the car to your own WiFi at home).