You can. But since almost nobody does, there's no viable use for a phone without it. Bear in mind, even Microsoft, who was part of starting this investigation against Google... Even Microsoft's apps like Outlook and Skype won't run without Play Services installed and enabled.

You can but a huge number of apps use Google Play Services and you can't use those. I suppose technically you couldn't get them anyway since you don't have Google Play, but anyway.

GPS includes stuff like location services, Firebase, Chromecast, analytics, Google account authentication, Google Drive APIs, all the game stuff, Android Pay, etc.

Essential code for apps written against the platform's official APIs.