Cygwin vim ships with vim's default configuration, which leaves vim in vi compatibility mode where it tries to emulate the original vi as closely as possible. Among other limitations, arrow keys do not work in that mode, and backspace just moves the cursor left rather than erasing a character.
Creating an empty
~/.vimrc is sufficient to disable vi compatibility mode:
Having said that, i to enter insert mode should work anyway. You'll need to provide more details on where and how you're running vim. Also, are you actually running the vim that comes with Cygwin, or the native Windows version of vim?
You can add below sets in ~/.vimrc to make is similar to default vim