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.

Yes, you can cast multiple spells at the same time and have both take effect

There are no rules against this

The rules are fairly clear about what is and is not allowed with regards to when and how spells can be cast. The restrictions on spellcasting timing come down to two major rule categories:

• Action economy
• Concentration

As long as you have the action available to cast the spell with all the appropriate rules that all spells must follow then you can cast it. There is no rule that restricts casting with respect to ongoing casting and certainly nothing that implies that it is in any way cancelled.

Sage Advice Compendium explicitly allows it and there appears to be no mistake

You quoted it and I understand being sceptical, but there is no reason to think that this is a mistake. It contradicts no rules and it is an official rules clarification document.

If counterspell did not work this way, there would be no way to counter a counterspell to one of your own spells. And the whole point of this Sage Advice was to clarify that counterspell is intended to allow this.

However, that is not the only example of casting spells at the same time.

Spells with casting times longer than 1 action/reaction is another place where this could happen

When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so.

Take this case for example:

• Cornelius begins casting simulacrum
• Cornelius casts feather fall in response to someone falling
• Cornelius continues casting simulacrum
• After the casting time is completed, simulacrum takes effect

In this case, you do actually kind of cast two spells at the same time. And this should work with any type of spell cast as a reaction during the casting of a spell with a duration greater than 1 action or reaction. As long as the caster maintains concentration and uses their action every turn to continue casting the long spell, they can do whatever they want.

Note that this will not work with bonus action spells because the restriction on those means that only 1 action cantrips can be cast the same turn.