First, it's mA, not mAh. The latter is milliAmp-hours, a unit of charge. The former is milliAmps, a unit of current.
If you want to use the switching regulator, just need to put a 330uF capacitor across the output and GND pins. Then it'll work fine. It'll probably be more efficient than the 7805, but don't expect miracles.
My guess is that it'd work fine down to about 7V input (for 5V output).
If your phone works without the battery, then sure, leave it out when using your external source.
The 1A used by the phone is typically only when charging the battery in the phone. If you're not charging, you'll likely be using less current, so your external battery would last longer. Exact durations depend on exact current draw and switcher efficiency.
My guess is that a car charger will be a switcher. It may work off of 9V as well. Or it may not. ;-) I'm helpful, huh?
As the other post says, your phone will also charge/run off of a USB port from a laptop, but in that case, you're using your laptop battery to run the phone. I don't know if that's desirable for you. Plus, most laptops shut off the USB ports when they're not on. That means you'd be running the laptop AND phone that way.
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