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"but a triangle cannot be split into anything else than triangles." is actually incorrect. A triangle can be converted into 3 quads. See computergraphics.stackexchange.com/a/9480/209
Just to add to Adam's excellent answer. WRT "This generally causes the result to be very noisy, and also gives a performance hit because of cache misses as those pixels are scattered all over the texture." If you use texture compression but then don't use MIP mapping, it'll undo the benefit of using texture compression in the first place! There are some notes here: computergraphics.stackexchange.com/a/419/209
The fact that the moon/planet doesn't look spherical to me seems to suggest that you might have perspective correct texturing turned on but haven't set the "W" (and I don't mean your width value) set correctly.
I believe Tom Forsyth (ex-game dev, ex-Larrabee engineer, etc) wrote a blog on using 64-bit ints for geometry, but my searching has failed me. You may have better luck.
There is a little bit of info on Texture Compression techniques on the S.E Graphics Community site but it doesn't cover ASTC as it is considerably more complicated than most other texture compression methods.
Are you sharing the vertices on the corners of the cube? (I'm just working this out in my head so might have this wrong but...) If so, given appropriate UVs you may get the right mapping for four of the faces, but the UVs for the end surfaces will give you the result you're seeing. You'll probably need to make those end squares use their own set of vertices.
[Disclaimer]I don't do IOS programming but I know a reasonable amount about PowerVR GPU's and also texture compression[/Disclaimer] Apologies for being lazy and not reading everything, but a) you aren't modifying a given texture from frame to frame are you? That might easily result in extra memory consumption as a TBDR GPU could be running asynchronously and the driver might want to avoid dependencies. b) If it's of help it seems to me the FIC system could be extended to use either 4bpp or 2bpp PVRTC instead of uncompressed data.
