When a nymph is killed, can items stolen by her just disappear

The most important thing to note is that identifying items in one game only helps you in that game. If ruby rings were rings of invisibility last game, but you died and started a new one, you know NOTHING about them. The exceptions are the trivial ones: potions of water and blank scrolls/spellbooks.

The second most important thing about identifying items is that each item can be blessed, uncursed, or cursed. If your role is Priest, then the blessed/uncursed/cursed status is automatically revealed to you for all items. If you are of any other role, then you need to utilize alternative methods to determine the status. The available options are:

• You can figure out which items are which by dropping them on an altar (as long as you are not blind, so that you can see the flashes of light).
• Reading a scroll of identify will also point out the status of the item.
• As mentioned in Durathor's answer, pets will avoid walking on cursed items that are on the ground, only moving "reluctantly" if they ever stand on it. You can use this from the start to avoid an early demise from using cursed equipment.

Blessed items and uncursed items are much safer to use than cursed ones. Don't drink any cursed potions, read cursed scrolls, or put on any cursed rings/armor unless you know exactly what you are doing.

Rings and armour

Conversely, if you have rings or armor that you know is not cursed, you can put it on and take it off right afterwards. For certain types of equipment (ring of invisibility, gauntlets of dexterity, boots of levitation, etc) this may identify it immediately. Others will have clear effects, allowing you to #name them yourself. (Ring of conflict, gloves of fumbling, etc) This will also tell you how heavily enchanted a piece of armor is.
Note that certain types of rare headgear can still cause problems, even if they are not cursed. The helm of opposite alignment will lose you all your divine protection, and will block you from doing the quest. Anything that looks like a "conical hat" should be avoided unless you are a wizard, in which case you may want to wear the expensive type.

Potions

Potions provide you with some interesting options.

• Clear potions are always water. However, they may be blessed (holy water) or cursed (unholy water), which have important uses. You can find out what kind of water you have by dropping the potion on an altar (they won't break, promise).
• You can try #dipping a unicorn horn into a potion. Most bad potions will be neutralized this way. If you dip a unicorn horn into a brown potion, and the brown potion clears/changes color, don't drink any brown potions.
  However, just because the potion did not change, does not mean it is safe to drink. Unicorn horns don't detect everything. (sleeping and acid, for example). Also, a potion of polymorph will cause you to lose your unicorn horn as it turns into something else. This can be prevented by dipping any other item into the potion first.
• If you find a food shop, you can try dropping your potions there. Anything that the shopkeeper offers to buy from you is a "food potion." In practice, that means water, fruit juice, or booze.
• Monsters will drink helpful potions (speed, healing, invisibility) and throw harmful potions (paralysis, blindness, acid) at you.
  When in doubt, wait for this to happen instead of drinking random potions. The penalties are much less severe, especially if the potion was cursed.
• If a potion was dropped by a nymph, and it's not one she stole from you, chances are that it's object detection.
• The potion of oil is unique in that you can apply it to light it on fire like a lamp. No other potion is a valid target to apply.

Scrolls

For scrolls, I typically end up relying on price. As with all items, more expensive scrolls do 'bigger' things. Some specifics on scrolls:

• First level of Sokoban always has two scrolls of earth in the same spot.
• If you find a scroll in a one-square "closet", it will be a scroll of teleportation.
• Scrolls of identify are really cheap. Cheaper than blank scrolls. This makes them pretty easy to identify.

For more information, including a very detailed price ID guide, see http://www.steelypips.org/nethack/id_faq.html

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.