If an Australian calls me mate, should I respond in the same fashion if I'm not an Australian?
For one thing, in Australia, "mate" runs the entire gamut of titles, from friendly (mate, oi mate, hey mate), to disappointed/incredulous (maaaate.) to hostile (Look, mate...) It's a one-size-fits-all title that pretty much replaces words like friend, acquaintance, pal, bud, buddy, etc.
As a Canadian expat (moved here to be with my Australian partner), I still struggle with this every time, because I weigh up the cost-benefit of
1) trying to fit in with;
2) the embarrassing feeling that me saying mate will sound ridiculous with my accent - and I'm sure to Aussies, it does.
What I've noticed, though, is that if you're actually in Australia, you won't ever be shamed for saying "mate", and I think just about every Australian seems to warm up to you as soon as you use it. So go ahead, mate *shudder* (it just doesn't sound right coming from me).
In a word, "No."
The word "mate" sounds natural when said with an Australian, English or Kiwi accent but sounds strange and fake when said with any other accent.
It's not "wrong" or disrespectful to call someone "mate" if they've already called you "mate" but if you speak English with a strong foreign accent, it comes across as trying too hard. Just leave it out.
Australians cringe when foreigners go around saying, "G'day, mate," in an Australian accent as though they are being clever and funny. But it's as silly as a foreigner in American going around saying "Howdy buddy," in a strong American accent. It's not funny. It's annoying and disrespectful.