Are the bridge holes pre drilled?

Actually, if you push the neck a bit to the left wouldn't that close the small gap up and also locate the strings a bit more centrally? Finishing on the neck will also add some thickness which will take up some of the gap, personally it doesn't look worth shimming.

Yes the nut looks too tall, but I'd wait until you are properly into setup before adjusting that. I would get everything else as close as possible, then set the nut height, then go back over and re-check relief/intonation etc. You will find that the strings will introduce quiet a bit more bow in the neck so you may actually have to wind the truss rod so it's slightly bent back wards when the strings are off to compensate. Again, something to be done during setup in small increments.