I'd put it above the picture rail. It would break up that horizontal solid line and also make your window look bigger. Be sure the rod is also wide enough that the curtains hang almost entirely off to the sides of the window, not blocking the glass.

I think I'd do a ceiling mount rather than drill into molding. If that's not possible, I'd just put it at the very top. I'd only go out about 5" each side on that size window.

I would go in between the crown and the ceiling. Because of the size of the crown you may need an extended rod to clear the crown. If you need light blocking you might have to add a blind as well. Keep in mind that unless you are using custom you'll have to work with available lengths. You can add a couple of inches of length if you use curtain rings.

I would go half way to beam with curtain panels. As for door covering. I find weird when people have panels on doors.