Unter, über, in, aus and außer are prepositions. Unter can also be an adverb meaning less than, but that's not the case here. The other words from your list are all adverbs.
You can use those adverbs to narrow down the meaning of the included preposition. For example unterhalb narrows down the meaning of the preposition unter:
Unter den metrischen Schrauben waren zöllige.
Among the metric bolts were imperial ones.
Below the metric bolts were imperial ones.
Unterhalb der metrischen Schrauben waren zöllige.
Below the metric bolts were imperial ones.
The adverbs oberhalb, innerhalb, außerhalb work very similar.
Es gibt oberhalb dieser Hütte keine weitere.
There's no other lodge above this one.
Er arbeitete innerhalb seiner Möglichkeiten.
He worked within his potential.
Er kommt von außerhalb der Stadt.
He comes from out of town.
The adverb hinunter is used with verbs of movement.
Sie bretterte den Skihang hinunter.
She hurtled down the ski slope.
The adverb herunter is pretty much the same, but the view is different: hin- means she's moving away from you, while her- means she's approaching you. That's the same for hinauf and herauf, hinein and herein, hinaus and heraus.
Geh hin!
Go there!
Komm her!
Come here!
The adverb rauf is a shortcut of herauf, rüber a shortcut of herüber, raus a shortcut for heraus. In some German dialects, nauf as a shortcut for hinauf, nüber as a shortcur of hinüber and naus as a shortcut of hinaus also exist.
The adverbs innen and außen mean something is on the inside or the outside of barrier, e.g. a wall or window.
Die Bedienung erfolgt von innen.
Operation is done from inside.
Die Scheiben sind außen total verdreckt.
The window panes are completely dirty on the outside.
The adverbs drinnen and draußen are used in a similar fashion, but it's not focusing on a barrier.
Die Kinder spielen draußen
The kids play outdoors.
Du bist in der Mannschaft drin(nen).
You are in the team.
Obenauf is a tricky word with mostly idiomatic use.
Er war wieder obenauf.
He was okay again.