There is a slight disconnect between what western languages and what the Japanese associate sincerity with.
This article about decoding asian business codewords explains the difference.
being Makoto (mah-koe-toe) means to properly discharge all of one's obligations so that every thing will flow smoothly and harmony will be maintained. It also means being careful not to say or do anything that would cause loss of face.
Increasing your ranking in the Sincerity skill allows you to maintain that face. An observer who sees through the sincerity would see the indecision and lack of inner harmony behind your statements. Failure at sincerity leads people to believe that your thoughts are incorrect - because they are disharmonious and have caused you loss of face.
A modern example would be a woman asking her boyfriend "Does my bum look big in this?". A man who says "You look wonderful." is being Makoto. His skill at Sincerity will determine how well that is received by his girlfriend. A man who says the opposite of what he really thinks is using deception.
In game terms, the first example is Sincerity (Honesty) opposed by Etiquette (Conversation). If the man fails loses to the girlfriend, the girlfriend thinks that he thinks the opposite and was trying to lie. If he beats her Etiquette roll, she accepts that that is what he thinks in the way it was intended.
Another example comes from internet-based argument. If side A has a factual and logical argument but makes the argument poorly or fails to connect with the emotional bias of side B, then he has rolled poorly on his Sincerity (Honesty). Side B continues to believe as he has been or believes that something is not the truth.
Define the Consequeces of Success and Failure Up Front
This answer addresses a very similar question. I think everything I said there applies equally here. In short: if you explicitly define the consequences of success and failure, players are less likely to misunderstand the information and run off doing some nonsense.
Let It Ride
In your case, there's one other trick I would add, shamelessly lifted from the RPG Burning Wheel:
A player shall test once against an obstacle and shall not roll again until conditions legitimately and drastically change. Neither GM nor player can call for a retest unless those conditions change. [The results of] the initial roll count for all applicable situations in play.
In other words, say you roll to search a room for stuff. You get an 8. That's it. That's your result for searching the room for stuff. If you retry the action, you don't reroll. If you try a new action, too bad, you're not gonna gain anything more. Likewise, if I roll 15 to climb a cliff, that applies to the whole climb; the GM can't ask me to reroll every 10 feet or anything stupid like that.
Let It Ride makes sure that rolls actually matter — you can't just turn a round an immediately invalidate something with another roll. Beyond that, it keeps the game moving forward. When we know that each and every result will stand, we can all focus on moving forward incorporating the result.
If you can't abide by a success or failure outcome, then don't put it on the table at all. Manufacturing excuses to reroll until you get the results you want is a sign that you need to rethink how you're scoping consequences. It's possible you shouldn't be asking for a roll at all.
If you want a situation to be a series of rolls, break it up into discrete tasks instead of just rolling amorphously a couple of times and then handwaving that, okay, now this one counts.
Call Them Out
If your players are constantly asking for rerolls, try just calling them out on their weaseling. Like, just straight-up say, "You're trying to weasel out of the outcome we already rolled for. Let's move on."
Why Are Your Rolling for "There is Nothing Here" Anyway?
If there's nothing to find, what's the roll about, anyway?
Occasionally there's some payoff to roll-to-find-out-if-you-know-that-nothing-is-here as a form of information-hiding, but from what I've seen, a lot of GM advice defaults to "Roll for everything just to create fake tension!" way, way too much.
In the example given, I'd only ask for a roll if I could frame it as something like one of these:
"If you succeed, you find everything of value in this room — secret passages, treasure, clues, everything." That way the PCs can discover something for their efforts even if it's not what they necessarily intended to find.
"Okay, so, time is of the essence, right? If you succeed, you find out right away whether there's a trap door here. If you fail, it'd take a long time to search properly." Now the roll is all about "What is the cost of the information you want?" I do this only when there is already pretty obvious pressure of some sort; otherwise you're just kinda manufacturing complications that don't really matter.
Otherwise I'd just tell 'em. There's very little down side to doing so. What's the point of trying to maintain a feeling of uncertainty here, unless you're trying to waste time on purpose?
Best Answer
1. How do I discourage players using the knowledge that they have a low roll to influence character decisions?
Be up front and honest with them about not Meta gaming. It is meta-gaming using the knowledge of a low roll to influence your in-character actions/reactions/thoughts/etc. I usually say just what you said, something along the lines of, "You think she seems to be telling the truth." If my player then goes on to think she is lying, I do one of two things:
The first option is always better, and usually in the long run makes for more engaging role playing when players can accept the cards their characters have been given. One major tip, when they roll low and you tell them they believe the NPC, try not to act devious or suspicious, your tone and demeanor can influence player thoughts tremendously.
2. Should I, and if so how, tell my players if somebody is actually telling the truth even if they have a very low roll to avoid them thinking they are being lied to and their characters are too stupid to figure it out?
This is a bit tougher... first I refer you back to the first part of my answer to the first question. Players acting on the knowledge of low or high rolls in a way that is discordant with the information you give them is Meta gaming. So, if the 2 options above don't work, then you have to ask yourself one question.
"How will having this player remain suspicious of the information affect the narrative?"
This situation is good for on the fly story development!
If the story really is better served with the character believing 100% that the NPC is not lying and there is no in game way to work around the suspicious nature of the player... as a last resort (and I personally would not do this), just tell them out of character that the NPC is being truthful. I can't think of a reason that you would need to have the character believing that the NPC was honest given the ad-libbed scenarios above, but if you find it imperative, it's your call to make.
...and nothing says the player won't suspect you of lying just because you are the DM. ;) Some players will always be suspicious no matter what as they will always believe the DM is "out to get them".
(And to quote another answer here after reading the stuff posted as I was typing this novel length answer... do be up front with your players that anything you say as a DM to the player will always be the truth. Lies might come from NPCs, but never the DM. And always make sure you are consistent in upholding these promises to your players.)