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.
A face can't be changed with silent image, but other options exist
Because the spell silent image is in the figment subschool of school of illusion, the spell "cannot make something seem to be something else." An illusion spell of glamer subschool like the spell disguise self, for example, can make a drow's face appear to be a different face, as spells of that subschool can make things "look, feel, taste, smell, or sound like something else."
However, the Disguise skill is usable untrained, and the drow's friends—if he has any—can aid him on the check… or, if a friend is more capable than the drow, the friend can make the Disguise skill check on the drow's behalf, disguising the drow. With enough friends—and the GM's permission—the drow should be able to roam the town unhindered until some high-level watchman with a super-high Perception modifier penetrates the disguise anyway. A disguise created by the Disguise skill has the added advantage of not being magical, so the spell detect magic won't reveal it, the spell dispel magic won't collapse it, and the spell true seeing won't penetrate it.
Also, as a level 5 PC—if this avenue is really important—, the drow could invest 1,800 gp of his 10,500 gp PC wealth in a hat of disguise. Just make sure the command word is utterly mundane so the drow doesn't look like a weirdo when he must reactivate the hat every 10 min.
Finally, confirm with the GM that in the GM's campaign that drow are still feared and hated. In many settings, drow are just different elves, and folks won't, like, kill them on sight or anything.
Best Answer
Cost
This is not without cost to the warlock; he has chosen to use one of his two invocations to get this thereby forgoing other choices. In addition, he uses an action to cast it and an action to move it; unless your battles are very static he would need to move it a lot. Remember, the most limited resource any creature has is not its spell slots or hit dice; it is its actions, it only gets one per turn. Good players know this and they should be thinking every turn "Is this the best thing I can do with my action right now?"
Innovation
This is a very clever and imaginative use of the spell - this is something that you should encourage in your players; not discourage. I have had a wizard use Silent Image to create an picture of a hallway that the rogue could walk behind; this not only gave advantage, it also allowed sneak attack against, coincidently, a bugbear.
Disadvantages
A 15 foot cube of fog rolling towards the bugbear is going to negate surprise (its just not natural) and allow it to make an active perception check to find out where the PCs are in the cloud. The bugbear can then use its action to attack (with disadvantage); following which it can see through the cloud because "Physical interaction with the image reveals it to be an illusion" - sticking your morning star into it qualifies as "physical interaction".
It doesn't work that way, anyhow
You say: " because the party knows it is an illusion, they are never hindered by it". Where does it say that in the rules?
The relevant text of the spell is:
There are only 2 ways to see through the image, "Physical interaction" or "use your action" and make a save. Knowing that it is an illusion doesn't allow you to see through it.