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.

A Glyph wards an area or an object. While the trigger can be refined to be conditionally dependent on (almost) anything you want it still probably necessarily needs to interact with the area or object you've warded:

You decide what triggers the glyph when you cast the spell. For glyphs inscribed on a surface, the most typical triggers include touching or standing on the glyph, removing another object covering the glyph, approaching within a certain distance of the glyph, or manipulating the object on which the glyph is inscribed. For glyphs inscribed within an object, the most common triggers include opening that object, approaching within a certain distance of the object, or seeing or reading the glyph. Once a glyph is triggered, this spell ends.

The above quote, preceeding the section on refining the trigger, does not indicate that the trigger can be anything but rather that it can be lots of things and also lists several things it definitely can be. If you go outside that list for the basic trigger, you need to confirm with your GM that the new trigger is possible. Unlike the open-ended section on refining triggers, it is not the case that the basic trigger can accomplish what you want.

Furthermore, there is historical precedence for it not doing so; Glyph of Warding in earlier editions of the game, while always very open ended, specified some version of the following clause (taken from AD&D 2.0):

A Glyph of Warding is a powerful inscription magically drawn to prevent unauthorized or hostile creatures from passing, entering, or opening.

which indicates the general scope of potential triggers-- they must in some way relate to the warded object or area. Refinement, however, is not nearly so limited.

This means the following death triggers are possible:

• This Glyph, placed above a doorframe, triggers upon any creature entering if Aroden is dead.

• This Glyph, placed upon the floor of the room, triggers if Aroden is killed while inside the warded area.

• This Glyph, placed upon the surface of Aroden's back, triggers upon his death

But the following triggers probably don't work:

• This Glyph, placed upon the wall of the study, triggers upon Aroden's death

• This permanent Glyph, placed upon the ceiling of the sitting room, creates a bright flash of light each time a creature dies.

In conclusion it's not proximity but interaction that matters, but proximity is usually a prerequisite of interaction. Non-local parts to trigger's refinement are fine, but the trigger itself (very probably) has to involve the warded object or area in some way.

Addendum for the specific purpose of dead-man-switching a WMD:

You can do this with the spell, and it is very appropriate. What you want is a glyph on each weapon that triggers "When a person not bearing the Royal Seal attempts to move, activate, or dispel the object, or if the object is moved, activated, or subjected to dispelling forces in any way and I am dead", and then you just tell the kingdom about the first part. You can't make the devices trigger upon your death but you can make them (and any other devices you feel so inclined to affect) work fine until you die and then turn upon their masters if they are ever used past that point. Basically, you are creating a cursed item whose curse's activation is delayed until your death.

This kind of warding, though, is still somewhat risky, since the glyph can be dispelled if the dispelling is done carefully, and then replaced with you none the wiser.

You could also ward yourself with a (surface, and thus mobile) Glyph that triggers when someone approaches or strikes the surface with the intent to kill you, and have that Glyph somehow remotely trigger the deadman operation via your choice of long-range communication spells and recievers, but you should be aware such a measure can be triggered against your wishes by a clever opponent of your home nation, or bypassed via clever assassination methods (e.g. poison for the specific wording I mentioned)

Basically, this does almost exactly what you want it to for your linked answer, but is a lot less infallible than you might have first thought.