Yes & Yes...
1. The M310's feature a sleep mode, after a period of non-use (bout 10~15mins or so) the mouse literally goes to sleep, similar in concept to laptop's.
2. There's an off / on switch on the bottom of the mouse so you can manually turn it off if you don't plan on using your computer for a while - going on vacation for example.
3. I'm using a M310 mouse with my laptop on a daily basis, never turn either one of them off. Got the mouse Aug~Sept 2016, after a year & half it's only on battery #2.
The M310's support Logitech's Unifying technology (but come with dumb receivers). Let's say you buy a Logitech wireless keyboard at a later date, if it comes with a unifying receiver you use it for both the mouse & keyboard. I'm running a Logitech K360 keyboard with the M310 mouse off one unifying receiver on my laptop. 6-months and no probs.
So happy with Logitech, picked up the MK530 keyboard / mouse set for a new tower build + another M310 mouse couple days ago.