NOTE: Im not dissing Paladins, just stating the obvious.

So, we have the A-Typical "Judge Dredd" Paladin, who thinks its his place to do everyone else's job... as in, he feels he need to check for traps, examine stone work, unlock chest and doors and dispense justice at his discretion because "I am the LAW!!!" literal quotes and narratives, while having NO skill sets or proficiencies outside of a lvl3 Paladin.

The party has conversations and plans to which he is included 100%, yet:

1. The party splits to circle around and see what lays in wait, half a round later the Paladin has double moved straight into the area in question with a blazing maul because he lacks dark vision. 
2. He will tie up vile necromancers burning bodies stating they need to be brought to justice, yet will cave in the skull of a LITERAL MESSENGER who happens to have correspondence concerning kidnapping...
3. Wants to stop and inspect EVERY unturned stone, broken piece of pottery and shrub in an area for loot...
4. Pushes past the rogue checking for traps and sets of a hallway of blender-esque blades that reduce him to 3hp, yet again, says he couldnt see ahead...
5. Loves to play mulligans after EVERY DAMN TURN, as in: as soon as the DM starts talking to the next player, said individual back tracks to add a spell, or bonus action, or a full on conversation with the DM while talking over the DM and other players... (This I addressed in game and nobody stopped me. I firmly let him know that this is like chess/checkers, once you take your hand off the piece your turn is done, you live with your actions until YOUR NEXT TURN. Its not fair to other players who sacrifice their time to play and its disrespectful to the DM.) 

So the DM tried something new. When stating we are going to search/explore and area, he makes us roll initiative and go in order, ONE AT A TIME. This way we avoid getting our asses handed to us (unless the P4 goes first), and nobody talks over another... even though "someone" seems to think they have a divine crystal ball and can see/hear everything the rest of us are doing.

Unnnngh, I shall name my first ulcer after this character.

This honestly seems like a player problem, rather than a character or class problem.  I think an OoC discussion is in order, followed by booting the player if they continue to monopolize the groups time and hog the spotlight

Yeah, I do not see how this has anything to do with the paladin class either. Just talk to the player out of session to give them a warning; tell them what their disruptive behaviors specifically are and request that they stop those behaviors. If they disrupt one more time, just boot them afterwards. If they they do not take the out of session talk well, you can boot them right then and there too.

Now that I'm not on mobile I can give a more indepth response: 

Quote from KangStew>>

NOTE: Im not dissing Paladins, just stating the obvious.

So, we have the A-Typical "Judge Dredd" Paladin, who thinks its his place to do everyone else's job... as in, he feels he need to check for traps, examine stone work, unlock chest and doors and dispense justice at his discretion because "I am the LAW!!!" literal quotes and narratives, while having NO skill sets or proficiencies outside of a lvl3 Paladin.

This sounds like a game monopolizer who doesn't really know how to play a paladin (and for the record, Paladin is one of the harder classes to role-play well, IMO)

The party has conversations and plans to which he is included 100%, yet:

1. The party splits to circle around and see what lays in wait, half a round later the Paladin has double moved straight into the area in question with a blazing maul because he lacks dark vision. 

D&D is a cooperative game.  If the player doesn't understand that, you (or better, the DM) should talk to them and help them understand that.  a PC Leroy Jenkins-ing the party's plans is probably ok, once, if in character, but remember, characters learn and grow too...or usually the party leaves them behind (especially if these moments almost get other PCs killed.)

1. He will tie up vile necromancers burning bodies stating they need to be brought to justice, yet will cave in the skull of a LITERAL MESSENGER who happens to have correspondence concerning kidnapping...

Sometimes mistakes happen, but if it is an ongoing pattern that is spoiling the tables fun, then again, its a "talk to the player" situation

1. Wants to stop and inspect EVERY unturned stone, broken piece of pottery and shrub in an area for loot...

The DM can control this to a certain extent, but again, a talk to the player OoC might be in order about what they can expect in the game.

1. Pushes past the rogue checking for traps and sets of a hallway of blender-esque blades that reduce him to 3hp, yet again, says he couldnt see ahead...

See "Leroy Jenkins-ing" above

1. Loves to play mulligans after EVERY DAMN TURN, as in: as soon as the DM starts talking to the next player, said individual back tracks to add a spell, or bonus action, or a full on conversation with the DM while talking over the DM and other players... (This I addressed in game and nobody stopped me. I firmly let him know that this is like chess/checkers, once you take your hand off the piece your turn is done, you live with your actions until YOUR NEXT TURN. Its not fair to other players who sacrifice their time to play and its disrespectful to the DM.) 

This is a huge problem, and one that should be addressed out of game, by the DM.  In game, if the player repeatedly tries this, the DM needs to shut it down.  If they still persist, it's probably time for them to find a new table.

So the DM tried something new. When stating we are going to search/explore and area, he makes us roll initiative and go in order, ONE AT A TIME. This way we avoid getting our asses handed to us (unless the P4 goes first), and nobody talks over another... even though "someone" seems to think they have a divine crystal ball and can see/hear everything the rest of us are doing.

If the DM is having to go to this extreme to manage this player, It really should be intervention/kick the player time.

Unnnngh, I shall name my first ulcer after this character.

I'd probably name it after the player.

This is a player problem. And forgetting the add ons. My solution to that is if I am 2 people pass that player, I don't count it. If I into the next round, I don't count and tell the player to know their pc. And If it happens a third time, I assign homework. 

Yes, yes indeed...

It IS an obvious player problem and wasn't highlighting the problem as a Paladin issue, hence the opening statement. This was more of a venting, more so than looking for insight. Though it is the GMs ultimate responsibility AND say-so, as they are the albeit supreme deity in their game, these issues have been addressed in and out of game. 

I was simply highlighting the issues and pointing out the GMs first step resolution in treating the player like a child ~ Initiative = Wait you turn. If it continues, the GM knows where we stand, but in the grand scheme of inclusiveness we are TRYING to work with him. I simply wanted to share the GMs "baby steps" approach.

For the sake of respect, we pm the GM after conferring with one another, but as that didnt seem to resonate with the player, I spoke out this last session (as stated above) and made it clear how the rest of the players felt. As expected, all we got was his obstinate rationalization for his actions, even though he is in on the plan from the get go. Going forward, if he barrels into everything blindly, we will let him and simply explore another area until he starts screaming for help... at which time we will respond, "Did you guys hear that?" or "Sorry, cant hear you from way over there... because the rest of the party STUCK TO THE PLAN" 

I will say there was a little bit of sweet justice last session: Said Paladin entered into a "game" with an undead fey where he had to entertain the creature with an item selection (50/50 chance item is beneficial or cursed), should we survive the game, we go free... so of course, the player selects the most obvious bait, a small statue of a suit of armor, thinking he is about to receive an upgrade. The statue was cursed and made his armor shrink until it broke away from his body into tiny pieces, leaving the Paladin in a half finished dungeon with a 7AC... effectively making him "hide" behind the rest of us because now he's afraid to just run into a room. 

Mwahahahaaaaa...(chokes a little from laughing) ...haaaaaa!!!!

I'm not really a fan of in-game shenanigans to deal with player issues. You may get some petty instant gratification out of them, but they rarely solve the issue. And I understand the initial baby steps kid gloved approach, everyone deserves a chance to do better, but if that fails it usually doesn't pay to stay that course much longer either. 

Talk to the player and when they say they are just roleplaying have them google the Wangrod Defense.

If the player is somewhat new then the DM  implementing the initiative order for exploration/social might be a good learning point for the player and can eventually be discarded once the player better understands the process. 

If the player is experienced and hasn’t played this way in the past with other characters but acting like this for this particular character then he may be trying to play in character but he needs to know he needs to pull it back for the other players sakes. And there should be in game consequences for his actions. 

and if he is experienced and always plays like this then he’s a wangrod and it might be time to kick him from the group. 

Assuming the player is experienced in D&D...

Let the player die due to their aggressive play style.  Seriously.  A traumatic event does wonders as a training opportunity.  Consider this your friendly attempt to properly train the player on how to play the game as a member of the group and within their class abilities.

1. Set a trap encounter that the rogue could have easily detected, then let the P4 do his usual.  Trap goes off and now you have one dead P4.  Alter the damage outcome to ensure it.  Let that kind of thing happen a few times and you may have encouraged the player to stop being stupid.  On the other hand, if after the 3,4,17th time, he may get the message and quit.

2. Does he talk out of turn while role playing?  Have it happen around a city guard captain or noble.  When they don't shut up have the captain/noble get insulting to the P4.  When the P4 continues to be an ass have them arrested and let them rot for a few days.  If they get belligerent and get aggressive with the guards over the arrest, well, now you have one dead P4.  Alter the damage outcome to ensure it.

3. Rinse and repeat until you achieve the desired outcome.  Player learns or quits.  Simple.

Had this issue with a party I was in.  Paladin did the same thing every time, no changes to routine.  Had a semi-intelligent sword he would ask questions of.

Him: "What's behind this door?"
Sword: "I have no idea. Never been here before."
Him: Kicks down door, trap triggers, party suffers.

repeat on next door.

So, the DM warned him a few times, out of the game then a few times in the game. For example, as he was praying to his god (Torm IIRC), he got the feeling that Torm was unhappy with him.  H didn't change, so the DM started giving him less XP, as he was not learning anything new.  He got the hint after a while and became a valuable character.