A basic knight-knave puzzle (for the purposes of this question) takes the form $Q(n, s, k, x)$, where:

$n$ men are guarding $n$ doors. The guards and doors are numbered for easy reference: Guard #1 guards Door #1, Guard #2 guards Door #2, and so on. $s$ doors leads to pots of gold (success), while the other $n-s$ lead to hungry lions (failure). The guards cannot move away from their assigned doors.

You know that $k$ guards always tell the truth (knights), and the other $n-k$ always lie (knaves).

You are allowed to ask $x$ questions in total – either interrogative or imperative. The questions do not have to be yes-no questions, and asking the same guard more than one question is permissible. A question can have no answer (e.g. asking a knight “point to a knight who isn’t yourself” when $k=1$), multiple answers (e.g. asking a knave “who is the knight?” and getting a list of all knaves), or cause the guard to respond “KABOOM!” for logical paradoxes (e.g. asking a guard “if this sentence is true, does Germany border China?”).

Under this notation, the classic case – 2 doors, 1 success, 1 knight, 1 knave, 1 question, “If I ask the other guard which door leads to the pot of gold, what would he say?” – would be $Q(2, 1, 1, 1)$.

(Some versions of the problem have guards present additional information, e.g. “Guard #1 says that Guard #2 is a knave”, or introduce “spies” who always answer randomly, etc., but these aspects are not considered relevant for the purposes of this question.)

What conditions must n, s, k, and x follow for a knight-knave puzzle $Q(n, s, k, x)$ to be possible, and impossible?

Trivially:

But what else?

Bonus questions:

What if you were only allowed to ask yes-no questions (in which case the 4 possible outputs are “yes”, “no”, “(none)”, and “KABOOM!” for logical paradoxes).

What if there was a limit $c$ (where $c<x$) on how many times you could ask the same guard a question?

If there are cases where a 100% guarantee is not possible and you are forced to make random guesses, are there at least optimal strategies to maximize your luck?

Would allowing guards to move away from doors or exchange doors with each other change the outcome of these knight-knave puzzles in some mind-screwy way I haven’t thought of?

What if we extend the premise to cases where $m$ men guard $n$ doors, and $m≠n$?