Can we also say “the landing gear doesn’t go down, get down, go up, come up, get up”? Are there any nuances?

Yes, there are several nuances. The preposition you pick will depend on the context of the subject. In this case, the landing gear, when flying, is supposed to be in the "up" position when flying and in the "down" position when landing. The big accident in Korea was caused by, along with several other factors, the landing gears not going down into the "down" position. If i were to say, "The landing gear didn't come up, so there was an accident." That would change the meaning to "The wheels did not retract into the body, so there was an accident." (Which is not the case)

go down, go up, and come up are all natural sounding, and can be used pretty interchangeably, get down/get up should not be used in this context

I can’t stop reading this post in James Brown’s voice For this context with a landing gear, however, I would say that “get up” and “get down” do not work at all. All the others are fine though

landing gear is lowered so you can land on it and raised to fly use up and down according to whether it is being raised or lowered. get down would only be used when addressing something or someone directly. Otherwise we would say get it down. or get the landing gear down

They're phrases that mean different things.

retract/extend These are literally the correct aviation terms. Retract the landing gear. Extend the landing gear. The landing gear won't/didn't extend. The landing gear won't/didn't retract. It isn't extending. It isn't retracting.

"Come" implies towards you (or a specific location, if one is named). If you are on the ground, looking up at an airplane, then the landing gear goes up and comes down. However, if you are on the airplane, with the landing gear beneath you, the landing gear comes up and goes down. It's a matter of perspective. "Gets up/down" sounds like you're speaking from the perspective of the landing gear itself.

Go down/go up does work but it sounds perhaps a little unprofessional/uneducated/childish (it's sort of the way someone who doesn't know the proper wording would say it). "Get" does not fit. "Come" works in both cases. You can also say "lower" and "retract" or "raise".

"Go" and "come" typically depend on where the event is happening and/or going in relation to you. I'd say if you're on the plane then the landing gear would be "going down" (away from you). If they were going the other way (towards you), then the wheels would be "coming up".

Theoretically we would use go down if we were on the plane since it would be moving away from us. and come down if we were not on the plane since it would in some sense be moving closer. In practice they are interchangeable. It's too abstracted to worry about.

I would say 'go down' is the most natural. The second and last, 'get down' and 'get up', give too much volition to the landing gear for it to be grammatical for me. The other two, 'go up' and 'come up', seem natural enough, but describe the opposite movement from 'go down'.

In this context come down works, alternatively you can use “deploy” in lieu of come down Get down doesn’t make sense, as getting down would be more used for people Get up is the same just in reverse Come up is also correct if you’re referring to the landing gear coming up after take off

"come up/down" has a more formal sound to my ear than "go up/down"