An illusion produced this way can only change to reflect its environment based on what the caster is aware of and only to the extent that they are aware of it.
An illusion of a mirror is only as good as the caster's perception of what should be seen in the face of the illusion (it's not a mirror, it is an image of a mirror)
The shadowy figure will only change to suit the lighting condition if the caster is aware of the change
This is also true for the the puddle showing rain drops
The torch does not illuminate anything. If the caster is unaware of the change in lighting conditions you will still see the torch even with everything else being totally black.
There are not many rules written on the matter of illusions, but what they do say is pretty clear and logically leads to the statements above.
Illusion spells deceive the senses or minds of others. They cause people to see things that are not there, to miss things that are there, to hear phantom noises, or to remember things that never happened.
The Minor Image spell (PHB p260) states:
You create a sound or an image of an object within range that lasts for the duration.
The image can’t create sound, light, smell, or any other sensory effect.
The rules say an illusion is not really there. It deceives the senses into experiencing a magical image of something that is not actually there. There is no light bouncing off anything, there is no light being produced by anything and we don't need to consider any physics (maybe a shame given I have a masters degree in physics). It's magic and does not obey physical rules. This is really important when considering the specific scenarios.
The Minor Image spell therefore creates a sound or an image of an object that is a deception of the senses and not really there. How does this all add up to explain what happens in the cases above?
- An illusion of a mirror when the scene before the mirror changes.
First of all the rules say it is NOT a mirror, it is an image of a mirror whose reflection appears as the caster wishes. It is only seen because the viewers' eyes are being deceived. There is no mirror, there is no actual reflection, no light involved.
If you shone a beam of light at it the beam would not bounce off anything, it would go straight on as there is nothing there. It won't make a little spot of light dance around the room unless it is within the area of effect, in this case a 5 foot cube, as part of the illusion.
For instance if the caster looks into their illusion of a mirror and has a smudge of soot on their face that they don't know about, the image they (or anyone else) see will not show the smudge of soot because the caster doesn't know about it. The caster will admittedly only see a faint image as they automatically save, but this is true never-the-less.
So to answer the question, the illusion of the mirror will show what the caster thinks it should show if it was a real mirror, no more and no less. There is no real reflection and it will not respond to environmental changes except through the perceptions of the caster. The magic takes care of artistic ability and imagination.
As a DM I would say someone looking into such an illusion of a mirror would get a save straight away under the interaction rule, unless the caster had a really good, detailed view of what a real mirror would be reflecting, of what the viewer should see. Getting advantage for this save from environmental factors would be relatively easy as the caster is likely to be significantly further from the mirror than the viewer if nothing else.
Does the caster have the ability to change the illusion this way at all? The sound version of the spell says this:
The sound continues unabated throughout the duration, or you can make discrete sounds at different times before the spell ends
Which means the caster can change the illusion as time passes. So yes.
All this makes illusions of mirrors produced this way easy to save against, and I think that should be the case as it is a very exact illusion. Of course it may be an illusion of a poor mirror, in which case the quality of the image of the reflection would not need to be so great.
- An illusion of a shadowy figure when the lighting changes
Most of the above is pertinent. If the caster is unaware of the change in the environment, i.e. in this case more or less light, then the illusion won't reflect the change as it is just an image created by the caster. Light does not bounce off it, it isn't really there, it is a deception of the senses. Shining a light on it when the caster cannot visualise the change would count as interaction and provoke a save. If the caster is aware of it the illusion can change to match the new environment, but otherwise the shadowy figure will remain the same shadowy figure even in the brightest light. This may lead to a save. DM's decision.
- An illusion of a puddle when it starts to rain
See above. Same argument.
- An illusion of a flaming torch when the lights go out
This is a tricky one. Following the logic above, the torch will still be seen, it is a deception of the senses and does not need light to be seen, but it will not illuminate anything as the illusion can produce no light. If the caster wanted to it could be an illusion of the torch and an illuminated 5 foot cube, the area of effect, but the light would not go outside this area and this would look really obvious (time for a save?). Changes to the real 5 foot cube environment (e.g. a rogue sneaking across it, a trip hazard introduced) would only be reflected in it if the caster was aware of it and wanted it as part of the illusion. Darkvision is no defence to the illusion as it is a deception of the senses and would effect that too, though it does mean that you would see the rogue sneaking across.
One way to think about illusions is to use modern concepts. I don't generally like making such comparisons as this is magic we are talking about, not technology, but the metaphor works well.
Consider you are wearing augmented reality glasses that allow a computer to project an image of something into your eyes as if it were really there i.e. you see it superimposed over the real world. Then someone shone a real torch at where you see the image. The image would not change, it would not reflect any light as it isn't really there. It's just an image, a deception of the senses tricking your eyes into seeing an image of something that isn't there. The only way the image would change is if the computer was aware of the light from the torch and changed the image to match, but it would not stop the light lighting up the wall behind it as it still passes straight through.
If the change was not made, or made badly, you would start suspecting it wasn't real... If you noticed there was no shadow and the torch light was illuminating the wall as if the image was not there, you might suspect something isn't right.
To complete the metaphor the computer is the caster and glasses are the magic spell. Your saving throw is thinking to take the glasses off. In real life doing this to people, even when they know it is an "illusion", you can scare people, make them try and pick things up or shy away from that venomous snake.