It is normally obvious, visually, where a ladder goes. You might say, "Why is that ladder there?" Of course, if the ladder goes to a hatch or trap door then you might use the verb "to go", but I'd normally refer to the hatch rather than the ladder.1) Where do these stairs lead to? There is nothing wrong with that question. You could also say "this staircase..." or "this flight of stairs ..."
2) Where do these stairs go to? Also good. But possibly better, "Where do these stairs go?"
Is the first one correct? Yes.
We use the same structure for ladders? No (although there might be exceptions).
As a linguistic purist (when convenient) I would prefer not to end the question with "to". Rather, "Where do these stairs lead?" and "Where do these stairs go"?1) Where do these stairs lead to?
2) Where do these stairs go to?
Is the first one correct?
Do we use the same structure for ladders?
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