Stood vs Stayed in a specific case

Solution 1:

1) These teachings stood untouched throughout generations.

2) These teachings stayed untouched throughout generations.

Neither one is idiomatic; throughout is incorrect with an indefinite temporal phrase like generations. If you really want to use throughout, you need a definite term like the middle ages or the sixteenth century. With an indefinite like generations, the prepositions should be for.

• 3) These teachings stood untouched for generations.
• 4) These teachings stayed untouched for generations.

Both (3) and (4) are grammatical. Of course, they don't mean the same precise thing, because they use different verbs with different meanings, and these verbs instantiate different metaphors.

• stood is the irregular past tense of the verb stand
• stayed is the regular past tense of the verb stay

Both stand and stay refer to continuation of the subject's physical location and orientation, so they can refer to the same phenomenon (though they don't always). But, in these examples, the subject (these teachings) is not a physical object, but an abstract reference, and therefore has no physical orientation to refer to.

This is a metaphor. And it fits both verbs. What is being said is that the teachings persisted historically (as teachings, part of the culture, whatever it is in this context) for (some unspecified number of) generations. And that persistence of an abstract idea can be metaphorically standing

• Democracy stands as a barrier to tyranny

or simply remaining

• Democracy stays firmly opposed to tyranny

depending on which ST- metaphor one chooses. There are a LOT of words with this phonosemantic constellation, and they come from a large assortment of ST-initial PIE roots, all having similar semantics.