On thermostats it's the difference between the off and on temperatures - like if you set it to 24 degrees, you could have a hysteresis of 1 degree, meaning it'll turn on again at 23 degrees. (Or something like that.)
Yes, but with a limited cycle life of typically 300 or so for AGM, significantly reduced if the battery is ever more than 50% discharged. Also high losses in charging, and needing an extended duration of high voltage low current charging to get to 100% from 80%
https://www.youtube.com/watch?v=-WqSGL_mc54