I can't answer the question on the G1W etc. cameras, but on connecting to the car battery I've given that some thought whether I want to or not. See the article
http://dashcamtalk.com/battery-discharge-prevention/ for a good discussion on how to do it and to use a battery discharge limiter to protect your battery from discharging too low.
Yes you can hard wire direct too the fuse box. I learned about these neat
add-a-circuit accessories reading about dashcams. You pull out a fuse to non-critical always-on circuit, plug in the add-a-circuit and pop in the fuse you took out into the add-a-circuit. So far you're back to the way it was. But you now have a place for a second fuse which gives you power off the wire lead. So there is your 12 volts.
Next you can get a cigarette lighter socket (or whatever they're called these days) from a car parts place, wire the socket to the new circuit and plug in your power to your camera. Now full-time power. Now to prevent your battery from draining completely, you can add a discharge limiter mentioned above that cuts power to the camera when the battery drops below X volts. On some the X is settable, on other devices it's a fixed voltage. Read about minimum voltages for car batteries, you want to stay above that, some cut out at too low a voltage by which time you've stressed the car battery and shortened its life.
With the above setup and if you get a camera that automatically switches to parking/motion detection mode, you are all set. Install and forget about it. As long as you have voltage > the discharge device cut off limit, you'll be recording driving or parking lot events. As for the battery in the camera, seems the issues are with hot weather, I'm not entirely sure. If you are in a hot area check out the discussions about batteries exploding in hot weather. I think you just leave the battery in the camera, some come out, some are soldered in. I haven't read about people pulling the battery out of the camera. If it's related to the G1W, I haven't read about that camera.
[Added] I just re-read your question, I'm not entirely sure but are you thinking when you run hard-wired you need to remove the battery? Whether running off the supplied power that comes with the camera or hardwired to the car, it's the same, the supplied charger is in the circuit, whether it's plugged into the cigarette lighter socket or the socket you add off the fuse box. The cameras won't run more than 10-20 minutes off the built-in battery so isn't a practical way to go. [end of added]
For videoing outside the car, my existing starter dash cam power comes by 5v mini usb plugged into the camera. I can unplug the camera (it'll turn off), remove the camera from the car, remember to turn it back on to record, and step out to video that plane crash, car crash, accident on road, on whatever awesome natural phenomenon is happening. Personally, I think it would be easier to just use my cellphone camera for videoing outside of the car. I would trade off video quality for ease of use of the cellphone over the dash cam. Remember also your dashcam will be behind your mirror and not so easy to access. Maybe your still running dashcam will provide another recording on what is going on outside.
At this point, I've read enough and learned enough from the starter camera and have started a conversation with an eBay seller to buy a Mini 0801. The trade-off is it doesn't have practical parking mode (it takes 8 button presses to get there in the menu). Also I'm not interested in wiring in a discharge limiter so parking mode is out. If I really need it, I'll use a $100 20,000 mAh lithium iPod charger with USB ports.
Ideal dashcam is a camera with variable OPTICAL zoom: wide, for large scenes and tele, for plate numbers. 
