Solutions for five faces, and some ideas for where to go from there.
With some assistance from @Stiv and @wimi.
7:9BOJ: Not solved
At the top we have a sun, a red dragon (from Mahjong) and the Greek letter Beta.
The numbers look like colour temperatures, but haven't got anything from them yet.
These are all frequencies of notes at standard concert pitch (all relative to A4=440Hz).
Translating them into note equivalents gives us:
A4 E6 B6
A5 D7 C7
B5 A4 F6
B5 A6 G5
Just looking at the letters, that doesn't give us anything meaningful.
But... we can assign letters to notes, starting with A4 = A and ignoring sharps/flats.
So A5=H, A6=O, A7=V, up to E7=Z:
4 A BC D EF G
5 H IJ K LM N
6 O PQ R ST U
7 V WX Y Z
And if we now translate our notes we get:
A S P
H Y X
I A T
I O N
So the answer to this face is: Asphyxiation
Time for some chemistry!
Replacing the various words with their symbol equivalents gives us:
Not sure what "TotallyRad" should be; but we have just two symbols, which makes Morse Code a possibility.
Taking O as - and H as . we get:
.- ??? A ??
.. --- I O
So TotallyRad is just Oxygen = - = T, and the answer to this face is: Dehydration
The numbers here are all elements.
Replacing them with their symbols we get:
S Ta Ru'
At I O
Taking the ' as an increment indicator changing the "u" to "v", we get the answer to this face: Starvation
The image posted here is inconsistent with the live puzzle, so taking the RGB values of the colours of the two squares from the live puzzle gives us:
R: DE 4E
G: F1 43
B: C1 59
The values from the first square can be read as leet-speak giving us "Defici". The second square isn't so straightforward, but all the hex values are in range for ASCII letters, giving us "ncy".
And with the "&" being taken as an E (possibly from its origin as a ligature of e and t), the answer for this face is: Deficiency
These look like co-ordinates, but translating to letters instead gives us:
And ordering those by the second letter gives us:
So the answer for this face is: Insomnia
Putting it all together:
Entering those into the interactive version of the puzzle doesn't do anything obvious, but...
The five faces solved so far are words for a lack of something:
Asphyxiation is a lack of air/oxygen, and air gave me a green light for that face.
Water works for the Dehydration face.
Sleep doesn't work for the Insomnia face, but dark does.
No obvious words for food or sustenance work for the Starvation face.
And a Deficiency could be a lack of a wide range of things.
Not sure where dark fits in, but Air and Water are two of the four classical elements (with Fire and Earth).
Trying those for some of the unsolved sides finds that Earth is the keyword for Starvation (not sure how that fits).
And Fire is the keyword for the unsolved Sun-Dragon-Beta face.
And looking again at the cube as a whole, we can see that Fire/Water are on opposite sides, as are Air/Earth. So maybe they form three pairs of opposites.
The Deficiency face is opposite the Insomnia/Dark face, and Light is the keyword that gets me the green light and a .COMPLETE. message from the live puzzle.
So in summary we have:
7:9BOJ: ??? Fire
7:41OC1: Asphyxiation Air
6:1EG: Dehydration Water
b21:3P2: Starvation Earth
61:7RP: Deficiency Light
1:2EG: Insomnia Dark
And I have no idea what the coded title of each face means.