I can't answer all your questions as you're doing stuff I've never tried. If your clay isn't hardening, though, I would suggest you leave it in the oven for longer.
It does depend on what clay you used - some are deliberately made to be flexible, you can even buy polymer clay that creates erasers, but the basic stuff should end up reasonably solid. If you're creating long thin poles or something like that then they will flex a bit, as will very flat thin slabs, but it should only be by a little. Perhaps there is something a bit odd about your oven?
Maybe a cold spot within it, or perhaps it's not heating as hot as it claims. Try different positions and experiment with leaving the clay in there for longer.
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