When I, for example, cast searing smite as a paladin, and then attack and hit successfully, does the spell end after this successful attack, or does the smite last until the concentration ends or the user ends it?
[RPG] Are Smite spells one-use only
dnd-5edurationpaladinspells
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Edit: This answer predates several rulings and clarifications made by WotC and Crawford in particular. I'm leaving it in place for historic purposes, but it's no longer a particularly useful answer.
Strictly speaking, there is no clear interpretation. All three cases are justifiable. Also note that 5e discourages literal "rules as written" meanings. As the designers have repeatedly said: "rulings, not rules." The rules were explicitly not written to be scrutinized as a lawyer scrutinizes the law, so we should not be surprised when the end result of "it's ambiguous" is what we find.
Firstly, "natural" melee weapons are, as far as I'm aware, considered melee weapon in 5e. [ See also.] There is no distinction between a mace and a hoof as far as "counts as a weapon" is concerned in 5e. I don't know if this is explicitly stated anywhere (I thought it was) but Unarmed Strike is explicitly listed as a weapon on the weapons table, and it's strongly implied since all monster stat blocks say things like "Bite Melee weapon attack: [...]". As far as I can tell, if you make an attack with it, it's considered a "weapon" in 5e. Something is a weapon if it's used to make an attack, then, not because it's got a weapon tag on it.
You could argue a Case 1 by saying that find steed only modifies the target of the spell. The spell still refers to "you," so even though it effects your mount, that extension does nothing. In other words, you argue that for Range: Self spells, "you" in the spell description means exactly, "you, the spell caster," and never, "you, the spell's target." This interpretation, however, also modifies spells like divine favor, detect evil and good, crusader's mantle (that one's a bit of a pickle to decode with a mount), and every other Range: Self spell. I'd argue it's all or nothing here. Either they all work on the mount (in some way), or none of them do. It doesn't matter how you rule here, but you should be consistent. Given the number and range of Paladin spells that are Range: Self I question an interpretation this narrow as being the design intent, but it's certainly supportable. About the only thing that reinforces this interpretation is the fact that the Smite class ability does not work on a find steed mount, but that's only because the class ability isn't a spell so it doesn't qualify for find steed's expansion.
The difference between Case 2 and Case 3 is deciding if find steed changes the wording of spells to "The first time both you and your mount hit with a melee weapon attack [...]" or changes to "The first time either you or your mount hit with a melee weapon attack [...]". Honestly, there's not enough information to decide either way. The spells are not written with find steed in mind, and find steed is not worded to make the end result clear.
You can argue Case 2 by saying, "The spell is intended to only affect a single melee attack; if it were intended to affect multiple targets, it would be higher level or otherwise deal less damage."
You can argue Case 3 by saying, "Find steed, like find familiar or hunter's mark, is a class ability masquerading as a 2nd level spell, and that wording was put there to have an intended effect. Furthermore, making a Paladin more deadly while mounted -- a fairly rare situation in most campaigns, IMX, and small Paladins are already less threatening -- is in-line with the desired result of the theme and flavor of the class. Given also the relative scarcity of spell slots, the additional power is probably not significant in most cases." This is not a particularly crunchy argument, but given that 5e does not separate crunch and fluff, it is legitimate.
If I were to rule conservatively, I would probably rule Case 2. If I were in a more liberal frame of mind, Case 3 would be reasonable. As it stands, I don't see any compelling justification for any one interpretation.
It means that you get an automatic +1d8 radiant damage from Improved Divine Smite, always, on all your melee attacks. You can also still voluntarily use Divine Smite itself to add +Xd8 radiant damage (depending as usual on how many spell slots you spend on it) to a chosen melee attack.
In other words, Improved Divine Smite doesn't replace Divine Smite and doesn't prevent you from using them together. That's all it's saying in the part you have emphasised.
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They only do extra damage on a single hit, but most of them have an ongoing effect for the spell's duration
Paladins have access to a total of 7 smite spells (all in the PHB):
Let's use searing smite as an example, as you've mentioned. Searing smite is a concentration spell with a duration of up to 1 minute. The spell description says:
The effect only ever applies to one creature - the first one you hit with a melee weapon attack while you're concentrating on the spell. However, searing smite has an ongoing effect that continues affecting that creature for the duration of the spell (though it specifies that if the creature passes the Con save or someone douses the flames, the spell ends as well).
All of the smite spells do some extra damage on a hit (specifically, the first hit with a melee weapon attack - or with any weapon attack, for banishing smite and branding smite - while concentrating on the spell). As noted in Eddymage's answer, an official ruling in the Sage Advice Compendium reiterates that this extra damage only occurs once, the next time you hit with a qualifying attack after you cast the spell.
Most of these "smite" spells - banishing smite, blinding smite, branding smite, searing smite, and wrathful smite - also have an ongoing effect that lasts the duration of the spell.
The other two "smite" spells have no ongoing effect. Staggering smite forces a Wisdom save; on a failed save, it gives the target disadvantage on attack rolls and ability checks, and the inability to take reactions, until the end of its next turn. Thunderous smite forces a Strength save to avoid being pushed 10 feet and knocked prone. The effect of the former has a set duration, while the effect of the latter just occurs instantaneously.