Perfectly OK, make sure the battery can supply enough current, which I suggest will be at least 1.5A
That's OK too.
That requires more explanation, buttons are generally wired between 0V and a digital input (most of the inputs labelled 'analogue' on most Arduinos are also digital inputs). Here is my take on connecting buttons: Buttons and other electro-mechanical inputs (introduction)
There are other tutorials here with other people's ideas, but connecting buttons between 0V and an input is standard practice, don't forget to enable the internal pull up resistors.
Yes.
It can be used to power the board too, as in as an input from a 5V power supply. Be aware that you should not put 5V on the 5V pin while the board is connected to your computer via the USB cable. As long as the 5V on the pin is within about 0.5V of the 5V PSU in your computer you will get away with having both connected, but one day...
You don't have to spend long reading the various topics on this forum to realise that Vin and the barrel jack are close to useless. Actually they are worse than useless because they fool beginners into thinking they are useful places to connect power to.