You should be able to do three panels in those windows up into that arch with horizontal louvers and fully functional. Your shutter maker just needs to line vertical posts up where you have vertical mullions! giving that center shutter something to hinge to. Here are a couple of similar windows I have done. It's a partial arch but it's the same concept with yours

Absolutely you can mix traditional windows with contemporary window treatments, although I don't think plantations are considered contemporary by any means. Ordinarily sash windows can have shutters installed over them, however I can see your sill is very narrow given your bottom pane sits forward of the frame. I can see privacy is no issue, in fact you have a great view and half height windows allow light in and keep peepers out, so I suggest plantation shutters fitted to your French doors, to allow tilt and directional light and a roman blind as the folds of fabric mimic the frames themselves to your corner arrangement. If they are too fussy for you a standard holland blind, however be creative, you can have a bonded holland which allows for a fabric front bonded to a block out backing, you can ask for a roman fold on the bottom making them look like a roman when rolled almost all up, and dress the window a little better, but Hollands can roll up into the frame and allow for maximum window when wanted.

Yes you can and in your case I'd go for a bi-fold plantation shutter that folds to the left and still gives you access to the cabinet on the right... e.g. it will look like the "left half" (if that makes sense) of this... with the top or bottom (or both) 'bi-folding to the left, e.g. like the top of this (slightly wider in your case)...

No be fine :)