I've applied for ETA through EasyETA.com. It says in the confirmation email that my application has been approved. However, I would like to double confirm my approval (as I did not apply through the official Australia immigration department immi.gov.au.
Is there a way for me to check it with the immigration department?
It seems best to go to the official Australia Department of Immigration and Border Protection Electronic Travel Authority web site to check your ETA.
It appears to be legitimate for third party web sites to accept applications for an ETA. It is noted by an official web site that: "Your travel agent or airline can also book your ETA."
