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.
Per Crawford, No: you can't use the paladin's Divine Smite with a thrown weapon.
The problem here, as you have correctly identified, is that the language is ambiguous: does "melee weapon attack" mean "a melee attack with a weapon" (your option 1), or "an attack with a melee weapon" (your option 2)?
Helpfully, rules designer Jeremy Crawford has answered this exact question in a tweet from August 2015 (which is not an official ruling), and in the following Sage Advice Compendium question on p10 of the v1.14 PDF (which is an official ruling):
What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon?
It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.
As such, we should interpret Divine Smite's trigger to mean "...when you hit a creature with a melee attack with a weapon...", which means that you can't use Divine Smite with a thrown weapon, as it isn't a melee attack.
Yes, Banishing Smite and Branding Smite can be used with ranged attacks, because ....
Spells are not limited based on similarly named spell mechanics
Despite sharing a similar name, Banishing Smite and Branding Smite are not actually related to the other smite spells or Smite ability of the Paladin class. So, this means that one only need to adhere to limitations present in the individual spell.
P.S. - The other smite spells (blinding, searing, staggering, thunderous, and wrathful) do actually require melee weapon attacks.