I can select Hangouts to replace SMS from the contacts app, but I don't see the option to use Hangouts instead of the default dialer. Is this option available without root?
This configuration, using the system's default contacts app, is officially unsupported as of now:
You won't be able to use Hangouts as the default app for calling on your Android device, like when you start calls in Google Contacts. But you can choose to use the Hangouts dialer when you touch a phone number on a website or other apps, like Google Search or Maps.
However, in lieu of rooting, you can instead try a third-party contacts application that broadcasts the proper intent for the Hangouts Dialer. Based on a quick test, it appears that Contacts+ does this, meaning that you can tap a contact and it will allow you to complete the call in Hangouts. Other third-party apps may support this as well.
You are probably missing the Hangouts Phone application.
Simply download it on the Play Store and you will have the Hangout dialer.