Electric assist is just that. You ride the bike just like any other bike and the assist comes in, usually in response to pedalling input. Some systems use twist or thumb grip throttles to give a boost when required.
It's a bit difficult to give a specific answer because it depends on the system and how much use you make of the assist, but you should be looking at a minimum of 50km between charges and many will go significantly further.
Look at the Airnimal Move.
The BionX system is an after market add-on that will work with virtually any bike. We have considerable experience with the system and would recommend that we fit it in our workshop, though it is quite possible for somebody with experience to install the system at home.
Most systems can be turned off and on, 'on the fly', so that the bike can be ridden just like an unassisted bike. Some, like the BionX add-on system, also feature variable assitance, again adjustable on the fly. Uniquely, the BionX also features a 'training' mode, where the motor works as a generator and provides additional resistance for the pedaller.
The UK does require that an electric assist bicycle is limited to 200W and 25kph (15mph). Of course, you can pedal much faster than this if you like, but the assist will stop at 25kph. So long as those limits are adhered to, the assisted bicycle is classified as a bicycle.
The assistance more than makes up for any weight disadvantage. They climb and accelerate faster than the same bike without an assist.
Most electric assist bikes have the same carrying systems as a conventional bike, so the same type of luggage carrying.