When a game is running in a web page, it can't enter full-screen automatically. You can only request full screen mode after any user input - touch or mouse event. For example "Touch On tap -> request full screen".
For your second question, you probably need to change the project scaling mode to Scale Inner or Scale Outer and enable scrolling (Unbounded scrollig=yes in layout properties).
Or make your own keyboard that doesn't mess with the screen: