I'll get right to it:

Is a single idea - one melody/one theme/one motif - enough to hold up a whole piece of classical piano? Can one piece last around 4 minutes with a single idea at it's core?

I ask because I've been told no - and every composor from before WW2 seems to be against me on this, yet I've played stuff upward 20 minutes long with a single chord progression (there's no melody) slowly modulating through the keys.

I mean this in the most practical way: Sure there's been loads of pieces about one "concept" like a whole opera about "love" or "war", but I mean something more technical: is an entire piece based around A minor, A flat Dim, C maj, E minor pushing it? At what point does that become pretentious, poppy, boring and some such?

Phonic

I can appreciate your hypothetical question, but that nature makes the question hard to answer. I've varied initial themes to the extent they were essentially unrecognizable. That's one way to build a larger piece out of a few ideas. I believe you would be best served by posting a recording that demonstrates what you do.

To address the four chord progression you mentioned (a min, g# dim, C, e min). It has some interesting aspects to it, It contrasts a diminished dominant function (g# dim) with a minor dominant chord (e min). There are many common tones between the last two chords (and the first). To make 4 monutes of this progression sound interesting you'd have to do some extraordinary things melodically or improvisationally. However, small harmonic shifts could take this to other interesting places. Why not show us what yhou could do with it.

Yeah, it's a tough one to answer, because a lot depends on how you go about it. Take Ravel's Bolero for example. It's just one single idea for the whole piece, and that goes on for at least 8-9 minutes depending on tempo. And yet, due to the subtleties of the orchestration, timbres, registrations, dynamics, etc. it actually works.

On the flip side, I've seen some pieces that are only 2 or 3 minutes long and they're already too long because of a shortage of material and ideas to sustain them.

If you are talking about building a 4-5 minute piece built only on a single 4-chord progression - I personally think that there's not enough there, but again, it's all in how you approach it. If you retain that chord progression, but do some interesting things melodically, texturally, rhythmically, etc. it could be done. Some pieces you might want to look into , if you haven't already, are Bach's passacaglia in c minor and his Goldberg variations, and the last movement of Brahms 4th symphony. These are excellent studies in what you can do using a single chord progression and yet still keep things fresh.