Learn English – A phrase for ‘something that happens rarely’

Here are a few commonplaces that might suit your needs:

Let's not get into that subject. It's a can of worms.

Don't pull that thread, or the whole thing will unravel.

Careful, it's a slippery slope.

There are a variety of words that can accomplish this goal. Most of them don't imply that the object can still move, but none of the ones I'm giving you imply that it can't. If you want to imply that it can still move, you could easily just add "right now", implying that at other times it may move.

• "Still" — This is one of its meanings, and probably the simplest word to use here.

• "Stationary" — This word means the same thing, though it sounds a bit more formal.

• "Stopped" — This one implies that it had been moving at some point, which for all of your examples is most likely true. It won't necessarily start moving again, though. This word can also be applied to things that aren't motion, but I think it works well.

• "Unmoving" — This just means not moving.

• "Motionless" — This one implies that there is no movement at all, so it might not be exactly what you're looking for.

And, as Au101 said,

• "at a standstill" — works perfectly as well, though a bit more wordy.