Alright, folks, it WAS one of the apps I downloaded, it appears. AllTrails. I still have to go back through and make sure of everything, but now the screen is the top user of power, not Android Services. So, maybe problem solved, for now.
S4 Mini clearly is running something (in the background, I think) that must be always "on" because it's sucking the battery down within a couple of hours, the phone is hot (not just warm, but uncomfortably hot), and slowed to a crawl. Charging is a sometimes thing--it's like being plugged in is...