Agreed that you should get them dry before you bring them in. Those temperature are higher than ideal, a little sip of water once a month may be needed to keep the roots of some cacti alive. Not enough water to induce growth, just enough to keep them from shriveling excessively. Some cacti are prone...

The way the roots are formed, you couldn't raise the caudex above the soil without also leaving a lot of roots exposed - I wasn't suggesting that! But I think there's a good chance that better soil is all it needs to perk it up.

It's normal for this species to have a caudex - though, according to Llifle, when grown from a cutting it usually fails to produce one. You got lucky! Kind of a rugged look is also normal. it doesn't look that bad. I don't think a heavy pruning is needed. I would just re-pot it in better soil, setti...

It's like somebody who wakes up after years in a coma.

There are a lot of random hybrids, too.

Bud set happens during dormancy, appears in the spring as an enlarged areole. If you don't see any of those, no chance of flowers this year (though they don't always develop the first year after the bud is set).

They like a little more water than most desert cacti, maybe you're not giving them enough.

Echinopsis hybrids are generally easy to grow.

The only way too much water would make them droop is if they rotted, which doesn't appear to be what's happening here. You've got two different cases: where the stem is bending over, and where the stem is still straight but it's leaning. The second case tells me you didn't firm the substrate down ad...

Agreed, that soil is staying wet much too long.

In the winter, they should be dry so they go dormant. You don't want them growing when the light levels are lower, and it sounds like the light is none to good at the best of times.

Maybe you should forget about growing them in pots and plant them in the ground outside, at least the roots would stay cooler.

The coolest place in the greenhouse is on the ground, maybe if you set them down there when it gets real hot it would help.

I'd suggest added shading during heat waves, but I see from your greenhouse thread you already thought of that.

I've found that cacti with weak root systems often will revive if they're under-potted (note that Lifle also suggests under-potting for this species). Problem is, it has a tap root so it needs a deep pot. My solution in such cases is to put rocks in the pot to take up the extra space, leaving a hole...

They're native to high mountain regions of Bolivia and Argentina, not adapted to high temperatures - that could be the problem. Might try moving them into the shade in hot weather. Put them in white pots to keep the roots cooler, too, and/or shade the pot.

Echinocactus platyacanthus? Probably wrong, but I took a shot.

When you bake it in the oven it takes a while, so the odor gets pretty strong (maybe cleaning the oven after would get rid of it, though). Microwaving is quicker, only a little odor which quickly dissipates. Another method I've seen suggested is to solarize it, put it in a plastic bag in the hot sun...