USS Missouri windows...

I assume that you are talking about the square bridge windows as opposed to the portholes.   Opening bridge windows is a matter of personal preference.  Often PE framing is applied and some black ink or paint is filled into the space.  It is your choice if you choose to do so. 

You can use Future floor polish to attach the PE to the clear plastic.   It has the extra benefit of helping to make the clear - more clear.   You may also apply the clear with Future or Elmers glue.  Paint the PE before applying, bend to shape, and touch up any paint nicks.  

The problem you may have when using clear plastic will be when you overcoat with the clear flat to kill the shine of the CA used to apply the PE.    

It depends on the scale as well. With a 1/700 model, I wouldn't bother trying to put "glass" behind the PE window frames because it will 1) Probably push the PE frames too far out and they will look funny and 2) I doubt it'd show up very well at that scale anyway. I just say the heck with it and use gloss black paint for square windows at that scale, and bore out the round portholes to give them some depth.

Now at 1/350 scale, I can kind of see doing that. I just know that trying to make the shiny spots of superglue go away after you get the parts assembled is one of the more frustrating chores of small ship modeling.