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:
Physical interaction with the image reveals it to be an
illusion, because things can pass through it. 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. If a
creature discerns the illusion for what it is, the creature
can see through the image.
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.
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
1. Can I create darkness with Major Image? Like a dark fog cloud that me and my group can hide in?
Major Image (PHB p258)
This is a visible phenomenon so yes you can.
2. If my team knows that it's an illusion, can they see through the darkness?
Major Image (PHB p258)
If each character discerns the illusion for what it is then yes they can. It is up to the DM to decide what discern means. If they are forced to figure it out for themselves it requires an action to examine it and they must succeed on a Int(investigation) check against the DC of the spell. If they are told what it is by the caster then the DM could rule they automatically succeed on this check or perhaps have to make it with with advantage as it they had been aided. If it is a standard tactic by the party and they know it is an illusion (as asked in the question) it may not even need an action as it they already have discerned it for what it is. However as already stated it is up to the DM to determine what discern means.
So yes if your team knows, has discerned it to be an illusion (as determined by the DM) they can see through it.
3. Do foes need to decide to make an Intelligence check in order to realize the illusion or they do the check anyway?
This paragraph assumes you have made an illusion of a physical object. However your "dark fog" is not physical, things are supposed to pass through it. The intent of this description is clearly to say that if a creature is given a reason to know that it is an illusion they are seeing, then they see through it. The fog described is much harder to discern compared to a brick wall and so would probably require the creature to examine it as detailed in the spell.
The spell specifically requires a creature to take an action to make an Int(Investigation) roll, passive Investigation is not enough. There is no solid component to the illusion to reveal it's non-existance physically and the illusion provides all the other required components:
So yes the foes have to decide to take an action examine the dark fog or be given some other strong reason to know it is an illusion (the first one to realise screams "it's an illusion!" for instance).
4. If I create a dark fog around my enemies, are they blinded? (as long as they don't know it's an illusion)
If the illusion is of something they cannot see through and they don't know it is an illusion then they cannot see through it. If the illusion was of a brick wall they would not be able to see what was on the other side. The same it true of this dark fog.
So yes they are blinded as long as the illusion is in effect and they are within it's area of effect. Foes outside the area of effect are not blinded of course, but cannot see into the area of effect.
It's a very potent way to use Major Image, but then it is 3rd level and a Fireball is pretty potent too.