battle.net – Does Battletag Change Affect the Numeric ID?

Online gameplay is not limited by where you play the game, no matter what country you connect to the internet from.

For Starcraft 2 specifically, the only thing that matters is what regional copy of the game you buy. Buying the NA version will connect you to the NA servers -- which may prove unbearably laggy from Australia, but won't stop you from playing it. See also: How does the StarCraft 2 region lock work?

It's either because your IP address changed (which is likely if you haven't specifically purchased a static IP plan from your internet service provider) or it's been a week since you last authenticated:

From the the Blizzard WoW Forums:

your account no less secure than it was before, the only change is that you will only be prompted once a week to enter unless you change IP address, from where you'll need to enter your authenticator code.

Unfortunately, the Blizzard FAQ is worded in a way that sounds like he new functionality won't always trigger

The Authenticator system will now intelligently track your login locations, and if you are logging in consistently from the same location, you may not be asked for an authenticator code. This change was made to make the authenticator process less intrusive when we are sure the person logging in to your account is you.

Also, according to the EU FAQs, this isn't active for SC2 accounts at this time:

Please note that this feature is not yet live for StarCraft II accounts.