Can you charge and use the battery gear while on the bike?
Depends on a couple things. Most of the stuff made by the "riding" companies, but not the "walking companies, will be able to connect to bike. Also, trying to charge the battery while heating the gear may draw more power than your bike's charging system can provide...which leads to burnt-out voltage regulator and no power to the bike or to you...and you can't ride very far on just the bike's battery...IME.
1. Has to be 12 volt gear; some of the older battery versions, and maybe some cheap new ones, ran on 7 volt;
2. Is it wired with proper connectors to hook up to the standard connectors you find on most heat controllers (see my 1st statement); is it wired internally to allow disconnection of battery when hooked to bike;
3. Those with self-battery have their own built-in heat controller, which has to have correct connectors to bike battery.
Probably the easiest approach to using heated gear on and off the bike is to buy regular bike-style gear (12 volt) with the heat controllers for use on the bike. Then buy a 12 volt Lithium-Ion battery (slightly smaller than a paperback book) with correct connectors for the gear. Fully charge the Lithium battery before going anywhere, and don't try to use or charge it while on the bike.
BTW, I use Gerbing and Warm-n-Safe gear and heat controllers. I've had good luck with both. I always use a dual controller so I can adjust both the gloves and the heated jacket separately; I always need more heat to hands than to body. Have two of the dual controllers, one for me and one for wife.
Here in Oregon, we get a lot of rain in the winter, but when we get a good cold front, the weather turns clear and cold, like mid teens and low 20s. That's when I use a 90 watt heated jacket (12 volts x 7.5 amps = 90 watts), 48w heated gloves, and 36w heated socks with an outer jacket and fleece-lined jeans. Any colder than that and I'd want the setup that JCZ listed above.