the first part
我是澳大利亚人 -- Correct and formal.
我是澳大利亚的 -- Correct and somewhat informal.
我是个澳大利亚人 -- Correct and formal.
我是个澳大利亚的人 -- Native speaker would understand you but it is weird.
我是一个澳大利亚人 -- Correct and formal.
我是一个澳大利亚的人 -- Native speaker would understand you but it is weird.
"我是(一)个澳大利亚的人" makes listener confusing by having a meaning of "Australia owns you".
the second part
All expressions are valid and can be used in most cases.
the third part
"我生在澳大利亚" is kind of weird but is acceptable, but "我在澳大利亚生了" is wrong. You'd better use "我在澳大利亚出生".
A single "生" generally means "give birth to/happen". Examples: "A生了B": "A gave birth to B". "无中生有": "Something happens/appears out of nowhere".
About 澳洲 and 澳大利亚
澳洲 can be treated as 澳大利亚 in most cases. But you should use 澳大利亚 to refer to Australia (country) and 澳洲 to Australia (continent) if needed.