So, it used to be that monster races had to be a certain faction.  I'm not sure if that is still the case, so that's my first question.

Second, the season 8 players guide mentions that to be part of any faction requires a background from SCAG.  So SCAG would have to be your +1 in order to have a faction.

Does that effectively mean I can't play a monster race from say, Volo's, since I would need SCAG for the faction and Volo's for the race itself?

You can choose background from any book and it does not affect your +1.    You can even create a custom one see pg 2 of Players guide.  With that being said they did reword it a little so one could interpret it as PHB+1 rule is now for backgrounds as well, so maybe they mean to not allow it.

Ironically the Dragonheist has lots of faction stuff and goes directly against the whole AL story that the factions fell apart.

Alright, I knew about custom backgrounds but wasn't sure about the background features themselves.