First I would look at a HD verti glide for the slider, if that doesn't do it for you, yes you can install the drapery rod higher. Maybe you install blinds in the big window, but its a bit large for many shades and blinds, with some drape side hangs. Look for ideas on Houzz.

1) Outside mount is more common than inside mount on patio doors, at least here in the midwest. You can mount it higher.

2) It's okay to do vertical blinds on the sliding door and horizontal on the regular windows, in fact, it's typically the best way of doing things.

3) I would stick to a horizontal on the window and vertical on the door. They naturally work this way anyways.

Like the above person stated, Hunter Douglas has a great line of products that come both horizontally and vertically so you can match products but just have different orientations. I'd check them out!