That's what I use. I've become a past master at standing up and breaking the link to the charger with two fingers, dropping the battery in my pocket and walking away.
Yes, they work away from the computer/charger (I'd actually recommend an external charger as there's a remote chance you could damage your computer). How long they work is dependent on the one you get. Roughly, 100 mAh is equivalent to 1 hour of vape time on average. I'm not average and chew through it faster than that--I get 10 hours or so out of the 1,300 mAh. I'd get around 4 hours out of the 650.
It's slightly better for the battery if you unplug it when it's fully charged and use it for a while before plugging in again. Lithium batteries don't particularly care to always be at full charge, it's a bit of a stress. I try to do that but I'm not all that great with it.