1: It will automatically run in full screen. On Android there is an option to display or not display the status bar, I assume IOS has the same option.
2: You will need to code your scripts specifically for touch input rather than mouse clicking. Stuff like Input.GetTouch[0]. That would be the input of the first touch, Input.GetTouch[1] would be the input of the second touch (as in second finger to simutaniously touch the screen)
3: You can install your app onto your phone without going through the app store if you are a developer (aka pay the $99 fee). You have to sign your application. The only way to void this is if your app is jailbroken.
switch your platform in Unity’s build settings to iOS . you can preview how your game will look like by selecting from the preset values for the aspect ratio in the game view window