You can do all those things if you want. Download the PDF to your laptop, add the pass to your phone's wallet, even print out a backup copy if you want. And then when you get to the airport, use whichever is most convenient for you.

I generally just get the mobile pass and put it in Google Wallet. If I ever run into a situation where my phone is lost/broken/battery-drained, I'll just go up to the check-in desk or a kiosk and get my boarding pass printed. No big deal.