Essentially, I'm trying to determine if there is a correlation between a drop in GDP per capita (e.g. during the 2008 financial crisis, Australia's GDP per capita dropped from USD \$49,000 to $42,000 2009) and the average income.
Regardless of the population size, what matters in terms of income is real or nominal terms. The gdp per capita may rise despite a constant population for the economy. But is it real terms that’s what matters. In an environment where the inflation skyrockets, gdp per capita is misleading.
Technically yes, but mind the fact that GDP per capita is, by definition, an arithmetic mean of a country's income; meaning that it measures directly the average economy's aggregate output per person in the country. 'Average income' can be interpreted in a number of ways, one of which is GDP-per-capita, but another is median income (50th percentile income), which can tell a very different story (particularly around recessions).
At face value, the question seems to answer itself since 'GDP-per-capita' and 'average income' are generally used interchangeably; however there is evidently some nuance here.
I hope I've answered your question.