1. you don't have to do medical test prior online lodgement, you can wait till you requested to do so by your CO. however, immi suggested that if you apply within 3 months of allocation in the website, you should get all docs ready for faster process of your application.
2. yes you will need idian police check. but again you can wait till you requested by your CO.
3. school uni letter, phone bill, bank letter, any letter sent to you with your name and address, i'm sure you have some. it would be weird if you live in australia with no letter sending to you.
all the best.