It depends on the system and GM style.

I usually would tell players the target number. Their character would typically have a sense of how hard something is, more so than a desk job nerd sitting comfortably at home trying to imagine climbing a brick wall. If I say climbing the wall is difficult enough they have slim odds, they can then make an informed choice.

DND is also largely missing meta game currency, degree of success, and succeed at a cost. All of those change how the game works, and make hidden rolls less appealing.

For stuff like "there's a hidden trap" or "they're lying to you", you don't want players to enter into meta game "I know there's something here so I'm going to be extra cautious" mode. I often find a hazard they can see and need to deal with is better than a hidden surprise. Like, all those black tiles shoot negative energy out when stepped on. And also a lot of Zombies just woke up and are shambling towards the tiles floor. Enjoy!

Personally I like how games like Fate you can mechanically reward players for going along with it. DND almost has that with Inspiration, but it's very under baked.

DND is also especially loosey-goosey about target numbers aside from physical combat defenses and damage.

Another system might have a more explicit "To bully your way past someone, roll your provoke vs their will" combined with "the bouncer's will score is 2". DND has vague rules no one uses for "asking a favor".

Sorry for a long unfocused answer. Happy to talk about whatever if you have questions

The book gives you examples of how DCs should translates to the world. Is this vaulting a head high wall, climbing a crumbling 2-story building, or scaling the outside of a tower in a storm? That need to know the number is only a problem when the table lets numbers replace story.

"You back up to get a running start and trip on a misplaced cobblestone just before you reach the wall." = you rolled a 2 and failed

"You latch into the crevices between bricks and skillfully clamber up until the window is within sight. There is only a one, last leap to make, when the brick beneath your anchor leg crumbles and gives way. You landed winded, but someone else might now chart a better route." = you rolled an 18 and only just failed

"You built as much speed as you could and manage to launch up against the rain-slick tower but your fingers fail to find any purchase, and you scrabble helplessly back to the ground." = You rolled a 19 and weren't even close to a success

Short version: No, not every game has target numbers.

Long version: within the d&d context, DCs set by the DM have only existed for about half of d&d's history, starting with 3e. Before that, "skill checks" worked in a few different ways including percentile dice compared to level, rolling under the most relevant ability score, or just rolling a d6. Outside the d&d context, my gut says that arbitrary target numbers are less common than fixed ones or ranges. In my experience, they're faster and simpler, too.

That's a valid mode of play, but I feel like if we're going to have agreed upon rules we should follow them, and not unilaterally change them. If the rules say "you spot the trap on a roll of 10 or above", the GM deciding you just don't spot it because they say so can feel wrong. It can feel like cheating. We had an agreement, and they just broke it.

On the other hand, if in your session 0 you all agree that the GM may fudge things for more drama, then have at it.

On the third hand, I've done things like "the rules say X but I think that's going to stink here. Anyone object to changing it?".

The important thing is everyone gives informed consent.

Shouldn't play a game with random rolls if one doesn't like random rolls. Secret rolls don't add anything except suspicion.

As a DM if I decide something is going to happen then I don't bother rolling. Like if a character who is competent wants to do something and they have plenty of time they just succeed. If a monster is sneaking I might just compare their stealth to character perception if being stealthy doesn't have more of an impact that the characters finding out they were being followed. If it has a game play impact then I roll openly but don't say what it is for. That way there is no suspicion that I rolled low and decided that it should just pass instead when the reason for the roll is eventually revealed.

Sure, sometimes. It should be used incredibly rarely. However, not in this way. The GM has plenty of levers to pull without messing with the one thing you have players for. If the GM is just going to tell a story then they should write a book. If they want to do cooperative storytelling then they need to cooperate.

If the rolls don't matter then the story gets incredibly boring, as it just goes whatever direction the GM wants. Without failure, success is boring. Without success, failure sucks. When they're perfectly balanced by the GM, it's predictable and not surprising or fun.

That's one way to play. Personally, if I knew the GM was secretly adjusting the game much I'd feel dissatisfied. Why not just give me a sticker that says "You win!" if I'm always going to win anyway?

Though this does tie into a separate bugbear of mine: D&D makes it hard to reason about encounters because the stats are unbound and all over the place. You see four bandits rummaging through the wagon they stole. Do each of them have 8 hp, 16 hp, 32 hp, 64 hp? Who knows! Do they attack once or twice? Could go either way! That is not an innate property of RPGs, but it's very common in D&D, and I think leads to a lot of "oh this is going badly - let me fudge the stats". Both because the GM got the math wrong, and because the players assumed these were 8 HP bandits and they're actually "well you're 5th level the bandits should be tougher" level scaling bandits.

I feel like I would never burn edge, but hold onto it like Elixirs in final fantasy. (Unless you can restore it somehow)

Depends on the system, style, and context.

For example, if I'm casting a spell on a victim in 5e, I know what the DC is.

For something like "finding the trap", in D&D that's pretty open to the GM. I usually tell players the target number before they roll, so they can better decide if they want to use more resources or rethink.

Other systems might have more specific rules.