Transfer the funds back to a bank account where you earn the income to avoid the hassle of thinking about this all together.
To answer your initial questions. You will pay income tax on income earned in the USA. If you receive interest payments on your Australian bank account, I imagine you'd be taxed by Australia on that interest, but if your bank account is over a certain threshold, Australia and the USA have data sharing treaties, meaning the IRS will know the details of your Australian bank account.
I think the whole guts of your questions comes down to the last one: "Will the country where I have the bank account think I'm evading taxes if I receive money but pay no taxes there?" Maybe. If you were a government agency, wouldn't accounts that fall into that scenario at least be a place to start analysis?
In the end, you're entitled to have bank accounts in various countries. If you're not doing anything dodgy, then don't worry about it. If you want to reduce the likely hood of even being swept-up as a "starting point" for analysis, then move the money back to the country where you earnt it.