I started the purifier and when I woke up in the Citadel two weeks later, Dogmeat was there (he was a companion at the time). But Fawkes wasn't (also a companion). Then I went to Underworld for the quest where I need under 50% health and Dogmeat disappeared. I went back to where they should be when fired but couldn't find either of them. What should I do?

For Fawkes: Go back to where you woke up. Companions don't always load in there correctly so leaving and returning makes them appear.
For Dogmeat: If he doesn't catch up with you when you fast-travel, try waiting 7 days until you get the message that he's gone to vault 101 and go pick him up there.

Thanks, hope this works!