No.
True Seeing says you can see through illusions, but doesn't say you cannot see illusions, just that you automatically can tell if what you see is an illusion or not. You could say that you automatically succeed your saving throw to disbelieve them, against most illusions anyway.
Lets check what happens when you succeed a saving throw against an illusion:
Saving Throws and Illusions (Disbelief)
Creatures encountering an illusion usually do not receive saving throws to recognize it as illusory until they study it carefully or interact with it in some fashion.
A successful saving throw against an illusion reveals it to be false, but a figment or phantasm remains as a translucent outline.
A failed saving throw indicates that a character fails to notice something is amiss. a character faced with proof that an illusion isn't real needs no saving throw. If any viewer successfully disbelieves an illusion and communicates this fact to others, each such viewer gains a saving throw with a +4 bonus.
Also, Shadow Conjurations are 20% real (made of shadow-stuff) thus even if illusions were invisible to you, that 20% of the summon monster would show up as a translucent monster that is partially real.
A shadow spell creates something that is partially real from extradimensional energy. Such illusions can have real effects. Damage dealt by a shadow illusion is real.
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
RAW I'd say no. A "visual" illusion spell cannot fool a True Seeing spell unless the illusion is more than 120 feet away and still visible at that range. (True Seeing has a range of 120 feet.) For big illusions that probably works at the outset but if the distance closes then the illusion falls apart visually.
Depending upon your acceptance of Psionics, there is a level 3 Telepath power called False Sensory Input (FSI) which you could target on a True Seeing creature to affect their vision (FSI explicitly states it fools True Seeing as it doesn't work on the target's eyes but rather their brain) then the combination of the illusion's sound, tactile, olfactory, and taste should completely counter True Seeing. The drawback being you'd need to also be a Psion Telepath class level 5. Optionally if you have Use Magical Device and can find a Psion Telepath to make you a dorje (wand) with False Sensory Input (FSI) on/in it, then you could use the "Use a Scroll" option to "fake" having access to the FSI power and combine the dorje with any illusion to mess up True Seeing folk right good and proper!