Filter Material Replacement

Foams (ie sponges) really only need squeezing in old tank water until they fall apart. But if yours have discoloured green and you want to replace it, cut both the old and new into four pieces and replace one portion at monthly intervals.

Brown discolouration is normal.

I don't know what the crystals are. What is the make and model of the filter?

Is it the filter that came with the tank? Sometimes they don't say what the filter is. In this case, what is the tank?

Looking at that link, it seems that the filter media includes cartridges made of sponge with carbon inside. If the crystals you mean are black, that's the carbon. You don't actually need it.
Carbon adsorbs chemicals, though not things like ammonia or nitrate. It gets full, and to keep on working has to be replaced. But if you replace the whole cartridge every time it gets full you won't be able to grow any bacteria on the sponge - carbon really needs replacing every couple of weeks.
The simplest thing would be to throw away the carbon and just use the sponge.

The plastic things and the cylindrical things are biomedia, which are the main home for the bacteria. The sponge part of the cartridge is biological and mechanical media - mechanical means it catches the debris.

If you want to replace the foam, I would think about replacing it with plain sponge rather than 'proper' cartridges. Though it is a bit hard to see from the link - is the carbon in a hole inside the sponge or just wrapped in it? If you want to use the proper foam, just throw away the carbon.

Carbon is a hangover from decades ago. Back then fishkeepers rarely did water changes as they thought they were dangerous for the fish. This lack of water changes caused the water to go yellow, and the fishkeepers didn't like that so they used a very simple filter to remove the yellow; basically just a box of carbon with a bit of floss on top to keep the bits out, and an air pump to pull the water through. As the nitrogen cycle became understood modern filters were developed to house bacteria, and the importance of water changes was also recognised. Carbon's only use now is to remove that yellow colour (though with a proper water change routine it should never develop), to remove medication once treatment has finished, and to remove the brown that leaches out of wood. But people get used to doing things a certain way and they still wanted their carbon so the manufacturers continued including it.
Carbon is cheap to make and it needs to be replaced regularly if used on a routine basis. The filter companies make a nice little profit from selling carbon so they won't let on that it isn't necessary.

You can always fill the space with more sponge. Or leave it empty in case you ever do need to use some carbon.