The image appears and moves naturally, otherwise it wouldn't be an illusion at all.

Pertinent text:

You can use your action to cause the image to move to any spot within range. As the image changes location, you can alter its appearance so that its movements appear natural for the image. For example, if you create an image of a creature and move it, you can alter the image so that it appears to be walking.

Another example of this would be creating the image of a spectral, headless horseman charging your target in eerie silence.

When you use your action to move it, the image moves as it would do so if it were natural, maintaining the illusion. After all, it would be incredibly easy to discern something was an illusion if it just disappeared and then reappeared every 5 feet. Since this spell requires an investigation check to discern if this is an illusion, the image by default has to appear real.

A static, non-moving fire does not appear real, and thereby would not be an illusion. So any interpretation that this is still image doesn't make any sense, and doesn't address the spell at all. At no point does the spell state that the image is static.

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.