I was wondering: how many of the regular platonic forms have been made with corner-, edge- *and* face-turning capabilities combined? Which ones? What are they called? Who made them? Which ones (if any) are still missing? Can they be made? Who will make them?

They only puzzle that comes to mind is Eric Vergo's Ultra-X viewtopic.php?t=19090 It has hexhedral, octahedral, and rhombic dodecahedral axes systems. But it doesn't strictly fit your requirements of being a platonic solid because it's a cuboctahedron not a cube or octahedron so it's a shape mod of what you want.

If you do want more shape mods, I suppose you could include a couple of the Dayan Gems, Clay n Eva's Concept 11, and Eitan's Comboctahedron as being shape mods of a tetrahedron.
Tetrahedrons have the advantage that face and vertex turns are basically the same so they should be easier to make. I can't think of any non shape mods that fit the requirements, but that could just be my memory failing me.

Wow, so what's the answer to my last question?

The Alphaminx through Epsilominx will satisfy all of the CEF-turning platonics. However, they are shape-mods.

To answer your last question, I might make them all... (I might be sick of the series by the Gammaminx, so we'll see. )

By my calculations, there are at least 80 distinct CEFT Cubes assuming a combination of puzzles that have only one type of cut(2 FT Cubes * 5 CT Cubes * 8 ET Cubes). The number is probably even greater, since combinations are likely more sensitive to cut-depth differences.

For example, the 3*3*3 can have anything from infinitesimal outer layers an a huge center slice to huge outer layers and an infinitesimal center slice(zero center slice makes it a 2*2*2). By itself, this variation in cut depth for the 3*3*3 is only cosmetic. However, a Helicopter Cube + Skewb + 3*3*3 probably has several variations based on the 3*3*3's cut depth.

For the record, I based my counts for CT and ET Cubes on wwwmwww's animations in this thread, which also do an excellent job of illustrating how cut depth effects the resulting puzzle. I am also counting the 2*2*2 and 3*3*3 has distinct FT cubes.

The full series of CEFT Dedecahedrons would be even more difficult to enumerate.

I wanted to answer something similar to Jeffrey but he saved my time. Thank you.
But I want to add something. Your calculation estimates only the number of puzzles when only plain cuts are allowed. With round cuts the number increases.

Before someone makes a CEF dodecahedron the CE-dodecahedron has to be made because that one is missing too.

I believe it's going to be more complicated than that. For example, there's at least 4 ways to combine a Compy cube and a 3x3.