Death by many Magic Missiles

Let's start with when to make Death Saving Throws (PHB, 197)

If you take any damage while you have 0 hit points, you suffer a death saving throw failure.

So any time something gives you damage while at 0 hit points, you suffer a saving throw failure.

As quoted above in the question, the language for Magic Missile is separate per missile. Each missile strikes simultaneously, but the damage from each is separate. If we are separating damage, we are separating those Saving Throw failures.

The Concentration function is similar in that it triggers off of damage (PHB, 203):

Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw.

However, it also goes on to state:

If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.

Jeremy Crawford specifically ruled on Concentration saves and Magic Missile requiring each missile to force a new save.

Roll for each missile

The real question

That leaves us the real question here as to whether or not Magic Missiles are separate sources of damage, or if the missiles hitting concurrently represent a single taking of damage.

Given that Crawford seems to believe that magic missiles are multiple sources that require a roll per source, it seems you can extrapolate that the missiles are also giving damage separately (while hitting concurrently) and thus forcing death save failures for each magic missile. The concurrent nature of the strike doesn't override the multiple deliveries of damage from separate sources.

Caveat Emptor

If you are a DM and planning to target a PC like this, you must beware of hard feelings at the table. Typically death saves are a PC thing, and if you are going to remove that by going after unconscious players then the possibility of hurt feelings is real. Most monsters are more concerned about taking out other creatures who are alive rather than double-tapping any downed creatures. If there was a real reason by the NPC to do this, then that's different - but if you're just going after unconscious players because they're easy targets to completely kill you may suffer backlash from your table.

Addressing your concerns in order:

Spell located in PHB pg.276

Passive, in combat

The only way to check it in combat passively is to pass something through it. Silent Image does not resist this as the wording of the spell states:

Physical interaction with the image reveals it to be an illusion, because things can pass through it.

However, in combat a creature doesn't naturally assume something is an illusion and attempt to wave a hand through it or run into it's space. This is why it requires an active check, which will be covered later.

Passive, out of combat

If you try to step on an apparently real bridge and tumble through, you would passively know that it was an illusion. This is covered much the same as above, in combat.

Active, in combat

This is the way the spell is actually worded. Specifically:

A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC.

This means that they have to use active checks in order to discern the image is an illusion if they're in combat. The reasoning behind this would be simple logic. If you were engaged in combat and an Earth Elemental suddenly rose from the ground, how certain are you that it's an illusion and not a spell? What are the consequences for being completely wrong if you tried to just ignore it and move into it's space? An active check is based on somebody thinking it's not real, and then testing it to make sure, but testing it relatively safely.

Consider that testing it may include prodding it with a sword or tossing a rock through it to check to see if it has substance. Basically, something that requires an action to attack, follow through, and wait to see the result. It's not a bad use of an action as it effectively defeats a hazard in a mere 6 seconds (so long as you pass the DC check!)

Active, out of combat

These are your typical take 20 scenarios. Any active check out of combat is something players can simply repeat over and over to ensure that what they're looking at is very likely to be real. However since the spell states that physical interaction reveals the illusion for Silent Image, the basics of an active investigation should be touching it with something, which would reveal it's nature.

Lastly:

You are correct: Any physical interaction sees through Silent Image. Be careful if utilizing this knowledge in game (meta-gaming) because as a DM I would start mixing in invisible creatures suddenly popping up and giving them opportunity attacks if you try to casually interact with them physically in the middle of combat. Reaching out to see if that Drow Warrior is really real is a great way to lose a hand or arm.