So, I have never played through the KOTFE content before, and this has been quite enjoyable. My question is this. What impact will my Alliance have if I decide not to complete the quest lines for obtaining people like Major Pierce, or the Doctor that wants all the stuff from the Rhakgul event?

 

Can I safely skip these?

Yeah companion recruitment from alerts isn't really necessary. You can always go back and do them later.

Yes. All the Alerts are optional. They are the means to bring back the Class Companions, your own for the particular class you play at the moment plus all the others. Not all Class Companions are available, but they come back in later stories after Eternal Throne. There are exceptions. Vik never returns as you learn in story he refuses to join you whether you're playing Trooper or not. Zenith has not come back at all, much to people's complaints for years. Jaessa I think can only return if you're playing Sith Warrior in a later story you haven't reached yet. It's after you complete Eternal Throne.

 

if a Companion never returns in a story you do not have to recruit them by the Alerts.