TL;DR: Give players one roll that encompasses all of their tries. If they fail, they have to come up with another solution. Solves a lot of problems and is more fun.
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The problem with doors
I have thought long and hard on this issue, like most of you. And I have read The Angry DM's article on only allowing a roll if there is a chance of failure and a penalty associated. And that's great advice, but the way most people implement it is flawed, in my view. Let me explain.
Do you know what the penalty is in my games when a character rolls a lock-picking check and fails? The damn door won't open, that's what. That's the penalty. Even in an empty dungeon without wandering monsters, there's a penalty. You can't get into the room that way. I guess we're going to have to have the barbarian try to bust it down. That's roleplaying, and that's fun, in my estimation.
By the way, when's the last time you had to have your strongest character try to bust down the door because the rogue couldn't pick the lock? Never, that's when. Because you have the rogue just keep trying until she gets it open, and she eventually will. Sure you have decided to put in a penalty for failure and trying again, like extra time that it takes and a roll on the wandering monster check. But when you roll for a wandering monster and roll a 'no encounter,' to your players' perspective there was no penalty for the failed check. What's more, thinking in terms of "I have to design some penalties for taking more time to retry checks" puts more work on you as the DM than there needs to be. Even if there is some wandering monster encounter for taking time for an extra check, it ends up just being a time-killing penalty that doesn't do much to add to the story or make the role-playing very fun.
So what happens when you're in an empty dungeon area with no wandering monsters and no monsters on the other side of the locked door? You don't let the rogue roll, you just give her an automatic success? That makes the rogue feel like her trained skill is a little bit worthless. Could have had the fighter unlock the door too without a roll (he would roll a 20 eventually, right?), and he would have an automatic success.
Also, does it feel strange to you that a level 1 rogue can seem to unlock any chest or door she comes across? Sure maybe she has to try a few times, and maybe the group has to fight off a wandering monster before she gets it open. But she gets it open eventually, no matter what. ... At level 1. Because eventually she will roll a 20 and get that very difficult door open, at level 1.
You know what's more fun? "You rolled too low. I guess you are still a bit of a beginner at lock-picking." And then they might have to have Ogar the barbarian try to bust down the door instead, or find another way in. ... That's fun. Not some time-wasting random monsters until finally she inevitably gets the door open anyway. Ogar never, ever gets to bash down the door with his 18 strength. Poor Ogar.
So what to do instead, and resolving some problems:
Well, first off, the philosophy that you only roll if there's a chance for failure and a penalty for failure still is in play. But when you get to a locked door, there is an obstacle. If you fail a check at some attempt to bypass the obstacle, the obstacle still exists, so there's your penalty for failure -- the door's still locked. Come up with another idea, guys -- there's your roleplaying and a bit of fun.
So, as other people have advised here, you get one check. It doesn't represent how many times you attempt, necessarily. It represents whether your attempts have succeeded.
By the way, as an important conceptual side note, I don't think the DC you assign to a locked door should represent how difficult the lock is, not directly anyway. I think it should represent the difficulty class that your skill roll has to meet. But if you are a level 1 rogue and are facing a DC 18 lock, and you roll a 20, then the lock must not have been too difficult to open, because after all a level 1 rogue was able to get it open with her limited training. And if a level 15 rogue, who is exceptional at picking locks, comes across a DC 10 lock and rolls a 2, well that lock is a well-crafted, very difficult lock to pick.
So conceptually, the difficulty of the lock is not represented directly by the DC rating, it is represented by how great of a measure you succeed or how great of a measure you fail. As a DM, you don't really know how difficult a lock is until the roll is made. You only have indirect control over that by setting a high or low DC.
So one roll. You fail, you fail, unless there is some other influencing factor that changes things. One of those influencing factors, in the way I run it, is if a character with a higher proficiency (or higher attribute) wants to try.
So, let's say the rogue is in another room because he fell through a hole in the floor and has to find her way back to the group. The other members find a locked door. The bard says, "Well our rogue isn't here, but I also have a lock pick set. I'm not very good at it, but I can give it a go." So he tries and fails. Suddenly, the rogue walks in. "Hey guys! Miss me?"
"Yeah, we can't get this door open."
"Well, let me try."
If the rogue has a higher proficiency, then I let her make a roll. But if she doesn't have a higher proficiency, well she can't get the darned lock picked either. No need for a roll.
There are a couple of potential problems though that you might have already thought of.
First problem (a question, really): Can more than one person try the skill? In the story narrative, yeah they can all grab the lock pick set and give it a shot. But in terms of the mechanics for the abstract attempts, only one roll. If if it fails, then you can only roll again if there is a change in the influencing circumstances, including if someone who is more skilled at the task wants to try.
Second problem: Couldn't the players meta-game and always have the worst person at a skill try first, then incrementally let other players who are better try? No they can't, because you as the DM don't allow it. If they are all in the room together, then they get one check. This will make them want the most skilled person at the task do the roll. This will make players feel like their proficiency at skills really matter. If some unusual circumstance occurs, like the most skilled player is out of the room for some reason but shows up later, then the more skilled player can give it a shot and roll. Once they get it open, they can brag about how the other players just don't know much about the fine art of lock-picking.
What about bashing in a door? Can't all the characters try? You let Ogar the barbarian try, and if he fails, well the door seems to be a heavy-set door that just won't budge. Every weaker character will also fail, no need for a roll. But, if the paladin says, "hey, let me put my shoulder into it too and help you," then the paladin is helping Ogar, and Ogar gets to roll again with advantage.
Running checks this way helps to bring in fun and interesting role-playing details. The strong players get to try to bash down a door if the rogue can't pick the lock (which almost never happens if you pretty much guarantee the rogue will pick the lock with multiple tries, whether you incorporate penalties for failures or not). The wizard gets to use his knock spell. Did you find a chest that you just can't open? "Let's take the whole chest with us and try to use a crowbar on it back home."
It also keeps you as the DM from having to use a system of boring time-consuming penalties that you wouldn't have designed in the first place. And it keeps the level 1 rogue from being able to open every single lock she comes across. Guess what rogue, this one is just out of reach of your ability at present. It becomes more rewarding to level up and increase proficiency. Fewer and fewer locked doors are inaccessible to a higher level rogue. But you have to make sure that some doors and chests and whatnot, are inaccessible to lower-level rogues.
