Nethack has secret passages (blocking corridors) and secret doors (embedded in room walls). You must find them before you can pass through them. The canonical way of discovery is searching, but it's not the only one. For example, if you're sure there's an hidden door in a certain square, you can try kicking it. If you're lucky you'll find the door, break it and gain some bonus experience points -- IIRC.
That's all good and fun, but where should you search?
Stairs always happen in rooms. Thus, you should look for "holes" in the map where a room could fit. If there are dead end corridors in that general direction you might be on the right path!
Here's a real life scenario. I marked some educated guesses with question marks to illustrate my point.
----- ------- ???????
|...| |.....| ? ?
|....###### |......# ? 3 ?
|{..| #......|# ? ?
----------- |...-#######-------# ???????
|..........#####|...| # ?
|.........| #-...| # ?
|`......<.-#####----- # ?
----------- ############## # ?
? ? #### ?
????????? ??????? #--|---?
? ? ? ? @|....|?
? 2 ? ? 1 ? #|....|?
? ? ? ? #.....|
????????? ??????? |....|
------
1 looks the most likely: a continuation of a dead end corridor. 2 is also plausible, that large area down there is so likely to have at least two rooms. 3 looks more far fetched, but it's exactly where the hidden door was.
I hope this still gives you an idea of how to make reasonable guesses when looking for secret passages.
Oh, and remember to never ever hold a key when playing Nethack. One search is hardly going to be enough, so run the search command multiple times the right, safe way: 20s (020s if you have numpad on) will search the current position for 20 times only if it is safe enough to do so. Hunger, approaching monsters and the like will abort your search, unlike holding a key does.
The penalty for killing it is actually pretty small, only one point. Which is regained easily by killing normal monsters. I would not bother to much thinking about it.
Funny detail, killing a peaceful has a -1 alignment penalty. Killing a hobbit has a +6 alignment bonus. So net you gain points. With a big if. You only get the bonus if the hobbit is hostile. So you first need to make the hobbit hostile and not kill it in one hit.
Murdering peaceful humans does have a large penalty. You lose two points of luck and telepathy. Unless you are chaotic, which has no penalty.
If you are doing it for moral reasons, tricking the game into killing it is still breaking your moral code. Bad hacker! No ascension for you.
If this doesn't bother you, you might try using pets, or Traps. Magic doesn't work (and does not give a warning). Perhaps polymorph system shock might work. But not sure on that.
Best Answer
You have a couple of options here.
If you have the spell of knock or a wand of opening, you can unlock the door without angering the Watch.
If you can obtain a key and you haven't already been warned by the Watch, then you can apply it on the door to unlock it, and you should only get a warning.
As Sconibulus pointed out, you can just wait for the Watch to be out of sight and then kick down the door as you normally would. The Watch only goes after you if they see you do something they don't like.
Finally, you can use your wand of digging or a pickaxe to go down a floor, and if you do it when your kitten is next to you, then it will just follow you down as though you had gone down the stairs with it. If your kitten is too far away to follow, you can just return by the up-stair on that floor and kick the door open from the inside. This shouldn't present a problem with the watch because none of them can see you do it, and you'll be reunited with your kitten.