At the the DM’s discretion, an impenitent paladin might be
forced to abandon this class and adopt another, or perhaps
to take the Oathbreaker paladin option
I read this as an out-of-character game rule: if your paladin falls, that is, your DM forces you to abandon the paladin class,1 you are required, as a player, to choose either Oathbreaker or some new class for your character. Oathbreaker’s placement in the DMG implies that this option is explicitly available only if your DM offers it.
Either way, you definitely do not have the option of remaining as a (non-Oathbreaker) paladin after you have fallen. You “lose” your powers, of course, since you are no longer a (non-Oathbreaker) paladin, but they are immediately replaced by those of Oathbreaker or your new class, as appropriate.
And when you fall, you are not abandoned by your deity, so much as you have already been abandoned and that is why you have fallen.
- As with previous editions (which were far worse about this), I strongly recommend that the narrative for how, when, and why a paladin falls should be worked out between the DM and player to maximize its effect and significance, and limit arguments and hurt feelings over differing definitions and expectations for what “good” entails. 5e does massively improve this process by making it less about morality, and more about keeping a specific oath, and by ensuring the paladin can continue to contribute and play in the game on a metagame level by replacing his powers, but I still consider it important that a fall be done in concert with the player, rather than unilaterally as the rules suggest.
The spell's duration is Instantaneous, so the Steed exists despite the Paladin dying.
The description of Instantaneous spells (PHB, p. 203) states:
The spell harms, heals, creates, or alters a creature or an object in a way that can't be dispelled, because its magic exists only for an instant.
For Find Steed, you instantly create a spirit that assumes the form of a steed that now exists independently of you. If you die, the steed can be presumed to still be loyal to you, as the "bond" is only worded as "long-lasting" and not "life-long".
What happens after the paladin's death is up to the DM, but it might guard your corpse until itself dies, or carry your corpse to a safe location if able, among other things.
Best Answer
In the Player's Handbook, page 86, the following is said:
The new class features for Oathbreaker paladins are on the page 97 of the Dungeon Master's Guide. Notably, they require the Paladin to be both evil and on their third level or greater.
The fifth edition of Dungeons and Dragons is generally light on forcing conduct rules on characters, instead preferring a less constrained model more reliant on the GM making good calls. If a GM wants to "enforce" the oaths beyond letting the player follow them as they wish, they need to be careful so that the possible punishments won't come in the way of enjoying the game. Notably, the only two mechanical punishments outlined in the rules (changing the class and becoming an Oathbreaker) significantly alter a character's nature and I would recommend never using them unless the player of the Paladin is fully aware of the change they're about to trigger by choosing to transgress their oath without repentance.