Can I enter two different addresses? A physical address and a mailing address.

The tax return specifically asks for your mailing address. For the address on your tax return, you enter the mailing address where the IRS (or state) can contact you by mail if ever necessary.  If completing a Schedule C for your business you can enter the actual business address for the business location.

Other than that you cannot enter two different addresses in the 'Personal Info' section.

Some states are aggressively looking for additional revenue. While you are given a choice in TurboTax to supply a mailing address and TurboTax says it doesn't have to be your residence... some states are now going over federal returns looking for people with mailing addresses in their state. With no other supporting data they are then sending out letters saying "we think you are a resident so refile your taxes and pay us $X plus interest and fees."