So by sheer divine incredible coincidence; The store I was planning on applying for a job at on Tuesday (adst) just posted a vacancy.
Now I'm paranoid that someone is going to get the job before me, how likely is that? It's now Sunday so that gives a whole day for someone to potentially beat me to it. I'm kind of worried...

Apart from that:
Should I just walk in and say "G'day, I was wondering if I could apply for a job or leave my resume here?"
What should I expect from any interview I may receive?
"Customer service representative" is basically cashier, right?
What's the best time of day to ask for an application?

I'll update when I've applied.

In regards to applications just walk in whenever and ask for one. Some employees may not be able to hand them out so you might have to wait for a manager or assistant manager to be in the store to give you one but it all depends on the store. I prefer to get and give an application from the manager because then I know that they have it, but don't really worry about someone stealing the position in two days. It takes time to go through applications ans do interviews so no one is going to just "swoop" in and steal the opening. Unless they already have someone lined up for the position then you better hope your application is better than theirs.