Do mobs spawn outside the border in Minecraft?

Yes they do spawn outside of the border, they can also pass through the border so you won't really have an issue with mob spawns, they'll just wonder right on in unless you have a wall.

There's a Minecraft map called Captive Minecraft which utilises the border. It starts off with you on 1 block, for every achievement you get, the border expands by half a block. The point is to get every achievement in the game, as the border expands it unlocks new areas and chest. During night time, mobs constantly come in from outside of the barrier