Fresh water tank vs city water

Yes, if you connect to city water, your pump and fresh tank will be overridden and you'll use city water.

Its not a very sophisticated approach, but, yes.  The check valve is only there to make sure that your pump doesn't push water out the city water connection.  Your pump itself keeps city water from filling the fresh tank. 

If you have pressurized the water lines with city water, your pump won't activate.  I suppose that, if you have really low city water pressure, your pump may come on, drawing water from your fresh tank, to bring the lines to full pressure... but it'd be a rare occurrence; and of course, you'll have had to turn on the electrical switch to the pump as well.