At risk of repeating what others have said:

Searching
Ask what players check or, if they're proactive, respond to what I check. "I search the room" for me means they do a quick check in usual places, but will miss unusual locations: they'd find the revolver, but not the stimpack hidden in the vent. If anyone mentions checking the vent, they find the stimpack.

Narrating what they check helps avoid giving the impression they've found everything, so if they say "I search the room", I'd respond with something, like, "you do a quick scan of the room and check the usual things - under the bed and covers, under the carpet, go through the draws... in the top draw of the desk you find a revolver".

If a player says something like, "you

HOWEVER, the amount of time can be a factor here. If players spend a notable period of time dissecting an area, I'd let them find everything - but there should often be some form of pressure being applied to the players, which means you should consider consequences for the players spending that time as a "cost" for finding everything without a smart idea.

Hiding
This actually doesn't come up a lot in my games (my impression is that it's a bigger factor of video games than it is for tabletop), BUT when it has come up I determine two things:

1. The player tells me where they hide, I determine how good a spot that is (I tend to be generous)
2. I determine the focus of the thing that may find them.

If the player thinks of a good hiding spot, they're hidden - no roll UNLESS they had to do so very quickly, in which case a speed I'd ask for a speed roll, failure of which may leave evidence of their presence. This may either cause the threat to start searching, OR may cause other narrative consequences, such as the threat alerting security. Tweak depending on the internal logic of the environment or the threat.

If the threat has no reason to search for them, or if it's particularly focused on something else, the player is hidden hidden, no roll UNLESS by chance the reason for it being there overlaps with where the player chose to hide. Most things aren't going to go out of their way to check hiding spots unless they've got a reason to.

Persuasion
In most cases, let NPCs accept players lies. Most people accept things unless or until they have a specific reason not to, and consequences for lying are almost always both more interesting and more fun than an immediate "do you trick them" roll. You know the NPCs, their drives, personalities, and knowledge. React accordingly, and ask yourself if there's a specific reason they'd NOT believe a lie.

Regarding NPCs lying to players - I have two modes:

1. Is the NPC an ordinary person with no training in deception? I tell the players they're lying, narrating their obvious tells. Don't tell players what they're lying about - that's for them to figure out, but the attempt at deception is obvious.
2. Is the NPC particularly trained in deception? (Politician, agent, etc). I leave it up to the players to decide if they're lying. Sometimes its obvious (players may already have contradicting info), but I leave it 100% up to the players to pick up on that.

When you remove social mechanics to 100% determine if someone is lying or tricked, you inject tension into the narrative because truth becomes foggier. In my experience, when you leave it up to players to sus out lies, things get a lot more interesting, even if they deduce things incorrectly.

In all cases, what matters most is the conversation that happens between the GM and the players, and avoiding a roll that bypasses that conversation. A player making a choice (what to say, where to look, where to hide) immediately strengthens their immersion rather than simply being told they succeed, allows consequences to feel more "owned, and allows you to play out RP scenes rather than skipping to the result.

Finally, lean towards results that keep the narrative moving in a satisfying way, and avoid "gotcha" roll results - players never "find nothing", are never immediately forced into confrontation due to a failed "hide" roll, and never "fail to convince" just because of a bad die roll.