The situation you are describing is "search for traps". You might hope to discover that the trap is not present, but in real life, you would do the same things to search for a trap that you thought you would find and to search for one that you'd hope to not find. Similarly, in game, you use the same mechanic, with the same failure states (ie, you won't find the trap regardless of roll if the trap is not there, and you won't find the cat if your roll fails, regardless of whether or not it's there).
In the case of "search for a clear path through the traps", again, what you're actually looking for are signs of the traps themselves. If you fail, then it's likely because you missed a trap that was there, and you will "find" a path that has traps in it.
If you were able to perceive the signs of passage of the natives of the place, you might search for paths that were frequently used, but that requires that you be able to spot those indicators. The difficulty is going to be higher. Also, it wont' tell you about what traps are there - merely that the natives of the place use a certain path. If they're all particularly short goblins wearing cloth or leather, there's still a very real possibility that your paladin in platemail will set off something weight-dependent or get hit by something that would pass harmlessly above their heads. Likewise, they might be temporarily disabling along the way and re-enabling once past.
Other than that, though, when we're searching for "a safe path", what we really mean is a pathway that doesn't have traps or other dangers... which means we're actually searching for the traps or other dangers.
Of course, this is 5th ed, which actively welcomes and encourages houserules on things like this. If the DM wants to rule that a particularly bad perception failure in an empty room makes you think you see signs of a trap that isn't there, that's up to them. It seems like the sort of thing that could be entertaining if played right. I'd keep it rare, though. One or two false positives in trapfinding could work if done well, but having a great many seems likely to just slow things down with minimal payoff.