Proper practice is actually to put the sheets horizontally-this is stronger and is simply the right way to do it. If you have a thinner rip, this always goes in the middle-not at the top or bottom of a wall. It is also always good practice to never have a joint at a door or window return, always return it over the door/window by a foot or two. Just like drywall. Good strong building methods. Also-just like drywall or brickwork-always stagger your plywood joints, by at least one stud space.
I always put a quarter inch or so gap-about the width of a carpenters pencil-in between sheets. I wouldn't put them tightly together. You need plywood that is meant for sheathing, most of the time standard construction plywood is fine, tell them at the lumber yard what your needs are they will tell you what to buy. I am assuming by "exterior plywood" you mean the sheathing that covers your framing, that will be covered by housewrap and siding.
Use hot dipped galvanized, 6 inches around the perimeter of the sheet, 12 inches in the field.
Most codes will dictate using nails-not screws-for sheathing. I wouldn't use screws. I am assuming you are using 3/8's or half inch for this, as most wall sheathing is this size. Five eight or 3 quarter inch is generally used for subflooring/decks.
If you are confused about technique to install it, go look at any house being build, look at how they do their plywood. (some are better than others however)