I would say Y has to do it since Y does not see or know the danger involved in doing what you are suggesting. The only thing that I can see wrong here is that if Y somehow knows that what you are suggesting is 'eventually' dangerous to him.
I base this on the description of charm/dominate where it clearly states that the effected will perform any action that does not pose a threat to his own health. Literal copy of the RAW (I used the 3.5e rules here since they offer a nice example which the Pathfinder version does not) for charm person:
An affected creature never obeys suicidal or
obviously harmful orders, but a charmed
fighter, for example, might believe you if
you assured him that the only chance to
save your life is for him to hold back an
onrushing red dragon for “just a few
seconds.”
The suggestion that you made to have Y move to a certain square sure seems to be a lot less dangerous than what is described above, so if it works with charm it should work with dominate/diplomacy in my opinion.
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.
Best Answer
Often, but not always
Many ability checks in combat require Actions, and many Actions require ability checks. Unambiguous cases are spelled out in the section on Actions in Combat (PHB, p. 192-193). For example, the "Hide" or "Search" actions both require ability checks, and both take up your Action (unless you have a special feature which allows you to do them some other way).
However, not every ability check in combat requires your full Action. A prime example is grappling. (PHB, p. 195, bold added)
In this case, we have two ability checks which do not require a full Action. One on the part of the grappler, which requires the use of one of their attacks, and another on the part of the target of the grapple which requires no action, reaction, or movement. (Of course, the grappler had to use the Attack Action to be able to grapple at all, but the ability check to grapple did not use up the entire Action).
Another example is found in the spell Tsunami (PHB, p. 284, bold added):
If this check required an action, the spell's description would say so. Contrast this with the description found in the spell Wrathful Smite (PHB, p. 289, bold added):
Does it take an Action? DM decides
As a general guideline, it's heavily implied that more difficult activities require actions: this is especially stressed in "Other Activity On Your Turn" (PHB, p. 190, bold added):
And it's understandable that you would think that all activities that require ability checks require an Action. After all, the definition of ability checks states (PHB, p. 186, bold added)
But this definition states "when you use an action that could fail, it requires an ability check": the converse, "when you use an ability check, it requires an action," is not necessarily true. If you attempt something in combat which is defined as taking an action and requiring an ability check, then it definitely requires both. Otherwise, the sections on both "Other Activity On Your Turn" (PHB p. 190) and "Improvising an Action" (PHB p. 193) leave the decision up to the DM when to call for the use of an action, the use of an ability check, both, or neither.