You can enter the US up to 10 days before you start your job.
ou can stay for up to 10 days after you finish your job.
Yes. The E-3 visa is a multiple-entry visa, so provided you have not changed employers, or made any other changes to your E-3 status, you may travel outside the US and reenter on a valid, unexpired E-3 visa.
No. You cannot be either self-employer or an independent contractor while you are in the US on an E-3 visa. You must have a US employer as your sponsor.