In A:TLA we never see more of the Air Nomads than the monks living at the various Air Temples. Did other Air Nomads exist outside of the Temples in the world at large?
While this is never specifically addressed within the show, it is fairly safe to assume that no, not all Air Nomads are monks.
Avatar Wiki specifically discusses boys who are being trained to be monks. I could see this being open to a bit of interpretation, but I see this passage as dividing boys training to become monks versus those not:
Young boys to be trained as monks have shaven heads, while monk elders grow beards and mustaches; they had the longest and most prominent facial hair of any nation. Female airbenders do not fully shave their heads, just their foreheads so their arrow is visible.
The Air Nomads are primarily based on Buddhist and Tibetan cultures. In these cultures, monks are not permitted to marry or have a family.
And that's the biggest supporting argument. If monks cannot marry or have families, there is no way for the Air Nomads to survive as a group.
It can, and should be noted that while Tenzin lives on Air Temple island and is charged with protecting the Air Acolytes, who are monks and nuns, Tenzin himself is never identified as a monk, just an Air Bender.
Per the Avatar: The Last Airbender bible, which was partially posted here, yes. In Aang's time, all Air Nation people were Airbenders and Air Nomads. The bible mentions of the Air Nation:
- Air Nomads are the most spiritually enlightened population.
- They are non-violent.
Combining both these, it's implied pretty heavily that they're all monks.
Obviously, since the genocide of the Air Nation, things changed. Non-benders (i.e. the Air Acolytes) now work with Tenzin and his family to maintain the Air Nomad culture.