Adding to what has been said here (and it's all been good!)
DnD has a history and thus there are layers to the paladin. To understand it you need to understand what it was and not just what it is now.
In 1st edition the paladin was arguably the strongest class (assuming the paladin had a holy avenger). It was balanced by way of some pretty strong restrictions on role play and some hefty requirements re: ability score. And given that the default world was Greyhawk and not Faerun playing lawful good was a challenge since evil is more powerful than good in Greyhawk.
Paladins had a 10 foot circle of dispel magic surrounding them and protection from evil field that went from 10 to 30 feet. In 1st protection from evil prevented demons, devils, elementals and summoned creatures from directly striking someone within the area and ranged attacks had -2 penalty. When a spell made it through the dispel zone the paladin still got a +2 to savings throws. And the paladin's sword was +5 weapon meaning it could be used against major deities. A paladin had a reasonable chance to solo one of the demon princes. And the list of abilities goes on:
https://tablo.com/david-thomson/1st-ed-ad-and-d-the-paladinBut the Paladin had to keep a charisma of 17 -- which was the default dump stat of 1st -- and they took a vow of poverty that required that they give away all gold at the end of adventure. This also meant couldn't own more than 5 items. If the paladin committed an evil act they would fall and become an inferior form of fighter -- which was a form of insta death for the character because the toon was no longer worth playing. So upholding one's lawful good oath was a constant worry on the part of the human playing a paladin. The discussion of whether a paladin was culpable for acts taken while being charmed or possessed was a frequent point of discussion.
And that's still the paladin people think of when they use the word -- the archetype of the lawful good crusader. The avatar from the ultima series, Joan of Arc, El cid, Lancelot, etc.
2nd edition moved away from insta death like mechanics and (eventually) gave mechanisms for the paladin to atone for evil actions and loosened the poverty requirements. Now, 2nd ed was hot mess and you will some sources that say that fallen paladins are always fallen and others say the atonement spell and a quest will restore a paladin. BG2 had a redemption option for fallen paladins -- TSR / WotC was okay with it redemption became official.
https://adnd2e.fandom.com/wiki/PaladinAs other have said the character of Lord Soth captured the imagination of many and that gave rise to the blackguard class. Instead of being a weaker form of fighter a fallen paladin could become a champion of evil. In 3rd edition allowed paladins and blackguards with paladins still being lawful good but able to atone. Paladins who didn't atone had to choose a second class to progress. Which makes becoming a blackguard an obvious temptation -- imagine regaining all the power you lost . . .
In 5th you can be any alignment so it's sometimes difficult to disentangle the archetype from it's current manifestation.
Question 1.In 1 - 3 eds the answer would have been yes. Paladins would have been part of some rule-bound, hierarchical order. Now, as Niara says, it's up the DM.
Question 3. I think this is where the history comes in. 5e allows DMs to use concepts from earlier editions (and the first modules actually had suggestions on how to run them using earlier rulesets). So the short answer is "up to the DM" the long answer is depends on how much the player and DM are drawing on the archetype that 1st edition created. If the player doesn't care for the oath but just wants to goodies that come with the class the DM can decide that the player isn't really taking the role playing requirements for the class seriously and punish the player.
Question 4. Of course up to players and DM but I'd say "smite the oathbreaker"!
Question 5. I could imagine a worshiper of Eldath serving as a "peacekeeper", smiting all those who break the peace. Were I DM I would expect the player to always try to avoid violence and to minimize the impact. Could be done in all seriousness or with the same humor as the old restrictions on clerics using bladed weapons -- "we're sworn not to draw blood, so we bludgeon people instead"!
Holy macaroni - thank you *very much* for your elaborate response. This clears up so many questions - and will make it much easier for me to make a believable paladin whenever the class is available.
Just... One question to clear things up beyond a doubt: does "smite" translate in attack, or does "smite" in DnD not necessarily mean violence? I am wondering because of the last lines of your response.
Also, I have always assumed but by now I feel like maybe I should ask - paladins would remain lawful or neutral at worst (like LG, NG, LN, N etc), right? I'd have a hard time imagining a paladin as chaotic... But perhaps one could build a character - like a paladin steward / apprentice that is more chaotic leaning but being led into the more lawful side by their mentor (which is a "true" paladin)?
Yeah that looks right to me. Serving an evil god does not mean you are an Oath Breaker. Usually Oath of Conquest are the default Evil paladins. Oath of Vengeance is usually true neutral or neutral good. The rest of the Oath's are various goods. Its only Oath breakers that are horrible and awkward imo. As a player I'd suggest just choosing Oath of Conquest if you want to be evil.
Evil gods muddled things for awhile but Oath of Conquest fixed that in my opinion.
I can see that, though an Archetype I like is actually a good Oathbreaker (which would have to be in something that is not the Faerun setting) where for whatever reason said character has become disillusioned with their god or the gods in general and have abandoned their title as a Paladin of said god. They could be trying to figure out if they want to serve a different god or be simply a godhater who still wants to do good despite hating the good gods and still hates the evil gods. Alternatively they could be serving a new god as a cleric but the Oathbreaker parts could signify that they still spurned their old god or are being spurned by that old god and thus have some lasting effects.
Err the point of my rambling is that while Evil Deities muddle up how many perceive Oathbreakers, I have always seen them as an archetype that can be flexible even if the MM says, "An oathbreaker is a paladin who breaks their sacred oaths to pursue some dark ambition or serve an evil power. Whatever light burned in the paladin's heart been extinguished. Only darkness remains," because I often times see morality as a little subjective and skewed by perspective at points, and gods can sometimes have some very extreme perspectives that abandon any sense of people being complex with many not evil reasons to do evil acts.
tldr; Even though many see Oathbreakers different from how they were before, I have always seen them a bit more different.
You propose an interesting story. But since paladins seems a bit more... Complicated, lorewise, than most other classes I'll remain default cookie-cutter mr. Nice-lawful-knight-guy until I better grasp the concept as a whole. :] I will however definitely remember your suggestion in the future.
From my understanding:
1. Paladins serve concepts in the form of tenets. These concepts are also a part of certain god portfolio's. Sometimes they directly serve the applicable god. Sometimes they serve churches, temples, or religious sects. Most of the time they adventure due to tenets. Many join the Order of the Gauntlet so they don't have to wait for orders from a temple or blocked by a government to strike down evil. Looking up "Order of the Gauntlet" faction if you are looking for something specific. They protect the weak and strike down evil preferably before things get out of hand.
2. Depending on the tenets, its usually "strike down evil, protect good". A lot of evil roams free. Clean up an area and move on would be my guess. Picking a subclass means you completed your training and took your vows.
3. If you unintentionally broke your vow. You have to spend a long time depending on the severity of the crime seeking absolution. This can be facilitated by another of your order. Both cleric and paladin are qualified to help you, but has to be from your specific order. If you are devout enough you could do it yourself (Dm's choice).
If you did so intentionally and the crime is severe enough, it use to be that the god would curse you several times over and turn you into a Death Knight(powerful undead) and be labelled an Oath Breaker. Doomed to seek redemption for as long as the god says so. When or if you redeem yourself the god would claim your soul and you'd be welcome to their realm.
Nowadays you just lose all power and either you suck at life or another god picks you up. Oath Breaker devolved to be an evil paladin that simply serves something evil.
4. Loss of love, warmth, and a piece of your very soul. They are divine casters. The very essence flowed through them. Nature, life, light, etc depending on who or what you served. Quite heartbreaking. They'd be very bitter and always looking over their back for the inevitable retribution. You devoted your life and were rewarded, but you turned your back. They would know immediately and it would be quite painful but definitely not all physically either.
5. Yes. Pantheons may favor you including who you follow to give you room to likely do what you want. Also creatures from the plane your god lives in technically could favor you as well. Unlikely but a DM may have a say in that. Oath of Vengeance comes to mind as an example of a paladin that does not necessarily have to serve a god at all.
Alright, that makes perfect sense, thank you very much.

Paladin is a divine class, they gain there spells/abilities related to there deity. You can be a good or evil pally, fall from faith and transfer or redeem yourself. They can be guards, deputies, judges, lawmen, or whatever. Originally they were Lawful Good or Lawful Evil, now though, I don't really know. Depending on the deity you follow you could take elements of what they are or just follow them period. In 4e Forgotten Realms players handbook Doomguide (level 11 paragon class / prestige class), are worshipers of Kelemvor. They have a obsession to destroy undead and manipulators of souls/death (necromancers). But any divine class can gain this, so the line between paladin, cleric, or whatever blurs a bit.
In forgotten realms all adventures require a license if you live or visiting X city. X being a city that requires a license, 4e campaign guide gives Amn and Cormyr as examples. In Neverwinter Nights Online you work for the city, while Neverwinter Nights one, your a adventurer that answered a call to help with a plague.
The last question sounds more like a cleric but you do you, d&d is a pnp game to begin with, using imagination and creativity to give the game life. FR was created by Ed Greenwood but the novels and such isn't written by one person.
Hmm, I was not aware of that license, so that is gonna require some digging. Thank you for your post.