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.
Insight checks are non-magical.
The description only states that you're immune to magic that does those things. However, an insight check is not magical (PHB 178):
Your Wisdom (Insight) check decides whether you can determine the true intentions of a creature ... Doing so involves gleaning clues from body language, speech habits, and changes in mannerisms.
Thus, the ring does not protect you from ordinary, non-magical insight checks.
As for DCs, there aren't any specific rules as to what the DC should be. Given that the interaction is wholly nonmagical, I usually use a contested deception vs. insight check when a scenario like this arises.
Best Answer
They affect you as normal
None of the effects you mention do that, so the ring has no effect on them.