Probably Not, that's what higher level illusions are for
Your quote for Silent Image covers what you can do with it (emphasis mine):
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.
Let's go over your questions:
Will the flames flicker naturally? Or does he have to use his action so the flames keep moving? What happen if he doesn't use his action? Do the flames froze and become perfectly still?
In this case the object would remain the object and most likely be still. Whether or not a campfire would be an object will be up to the DM.
In order to have the Silent Image move RAW, it must move to a new spot within range. As the image moves to that spot, the appearance is altered to appear natural. You'd need to move the campfire around in order for the flames to flicker, etc.
When casting silent image to create the image of a campfire, can the caster make it looks like the fire is lighting up (starting from a spark then grow)
No. Again, the way for Silent Image to 'move' is to have shift locations and it will create a natural appearance of movement.
Campfire maybe, Light No.
As shown in this question Silent Image does actually produce light or simulate it. You may have an image of an object, but it's a static image that does not have "any other sensory effects."
Major Image has the additional language:
It seems completely real, including sounds, smells, and temperature appropriate to the thing depicted.
You could emulate terrain on fire with Hallucinatory Terrain and Mirage Arcane, which states:
You make natural terrain in a 150-foot cube in range look, sound, and smell like some other sort of natural terrain.
Silent Image can do a lot of things, but it is not a replacement for higher level illusion spells like Major Image, Hallucinatory Terrain, Programmed Illusion, Mirage Arcane, etc.
You may argue that light is light, but in this case, illusions do not work like. See the following questions for additional support: