The keyboard charging is separate from the iPad. In fact they use different cables. But you can use the same power adapter from the iPad since the keyboard comes with the cable but not a power adapter.
Both the keyboard and the iPad are charged independently. You will probably find that the keyboard needs to be charged much less frequently than your iPad.
Yes you have to charge both and no, one device does not charge the other, actually they use diferent charging devices. The one for the mini-ipad does not fit the logitech keyboard and vice versa.
The keyboard and iPad both need to be charged. The keyboard lasts for weeks or months while the iPad needs to be charged every couple days in my experience. They each have their own battery so each one needs to be charged. There is no power shared between the two devices.
Yes - you do need to charge it. I have a wall plug in for my Samsung galaxy phone that I can plug two items into and charge at the same time. However, the keyboard does not need to be charged very often mine does not see heavy use.