Protein in fish food becomes ammonia. It can be digested by fish or decomposed by microorganisms, but either way it becomes ammonia.
Ammonia also enters the tank in the tap water if your water company uses chloramine. Some well water also has ammonia.
Beneficial bacteria live on all the surfaces in the tank, but especially on the filter media.
They need high oxygen and a source of food. Ammonia is their food. They thrive in colonies that include many species of microorganisms, but I am just talking about the ones that handle ammonia.
These bacteria (a species similar to Nitrosomonas marina is the fresh water species) turn the ammonia into nitrite, essentially by adding oxygen to it. They release the NO2 into the water as waste.
Another species of bacteria (Nitrospira sp) use the nitrite as food. They add another oxygen and turn it into NO3. They release the NO3 into the water as waste.
When these bacteria are healthy, doing their job, and are present in the right numbers you should not see ammonia or nitrite in the water. You might see a bit of ammonia right after a water change if your water contains chloramine. Some tests will show that, even if your dechlorinator has locked it up. It should go away overnight or faster. The end result of these bacteria doing their job is rising nitrates.
When these bacteria are not up to the job you will see ammonia and/or nitrite.
Why these bacteria might not be up to the job:
1) They need carbonate and a few other minerals. If the water is too soft they do not work very well. KH needs to be at least 3 German degrees of hardness, and harder is better. (Not better for the discus, unfortunately)
2) This usually translates as a pH over 6.5, and higher is better for these bacteria. The problem is that when ammonia is present in the water it can be in either of 2 forms. When the pH is low, the ammonia is in the form of ammonium, NH4-. This is less toxic to the fish. When the pH is high, the ammonia is in the form of ammonia (NH3) and this is the more toxic form.
The catch is this: For the bacteria to grow they need ammonia and high pH.
But ammonia in high pH water is toxic to fish.
3) The bacteria might need more oxygen. A dirty filter that has a poor water flow will not bring the oxygen to all the parts of the filter, so the bacteria can start to die off in those areas. Discus thrive in warm water, but warm water holds less oxygen. You need to keep the filter clean so water will flow through all the areas. Old filter media might need to be thrown away, but you are also throwing away the beneficial bacteria. Right now, when you are having problems, I would simply rinse and reuse the media, even if it is falling apart. Rinse it in water removed from the tank for a water change.
What I would do in this situation:
1) Water changes. Make sure the new water has enough dechlor for the chlorine or chloramine the water company is adding, and a little extra for the ammonia that is already in the tank. Read the label on your dechlor. If it does not handle ammonia, then switch to Prime or another one that does. With each water change clean the floor of the tank. If it is densely planted you won't actually do a gravel vacuum, but get the vacuum close to the floor and among the plants to remove as much debris, fish poop and fallen food as possible. 50% daily is fine, and will really drop the NO3 pretty fast.
2) Clean the filter. On the day you clean the filter do not bother vacuuming the floor of the tank. Just siphon tank water into at least 3 buckets.
Take the filter apart and rinse the media until it is about 75% cleaner than it was. Do not get more aggressive than this. The way I do this is to gently slosh the hard media (Bio noodles for example) in the first bucket, then the second as the first gets dirty, then the 3rd as the second gets darker. Gently squeeze the sponges and floss, using the buckets as needed, dirty, a bit cleaner, and cleanest. By the time I am done with a long time neglected filter even the 3rd bucket can be black. If you want to use Activated carbon, then go ahead and replace this.
Repeat the filter cleaning in a month. At this time you can (if you want to) replace ONE of whatever floss or sponge is getting too matted to really filter very well. Do not replace all the media at one time. Sponges last for MANY years. It is very rare for me to throw away a sponge. The finer flosses do mat down and need replacing occasionally.
3) Monitor the ammonia, nitrite and nitrate levels. When there have been problems with ammonia and nitrite, then the bacteria recover they can produce more nitrate. (This is good- you did not want the ammonia or nitrite hanging around, did you? Well, it has got to go somewhere!) When you get a good feel for how long it takes the NO3 to rise between water changes, then you can figure out a schedule of water changes (volume and frequency) to keep the NO3 under control.
Plants with algae.
1) DO nothing. The algae is helping to get rid of ammonia. If you kill the algae then there is more ammonia (decaying algae) and less ammonia removing things.
2) As the nitrifying bacteria get better, then as bad as that anubias is, I would trim the algae, just like mowing the lawn. Cut it off as short as possible and vacuum away the trimmings. Then do this:
With the equipment (filter, other water moving things) off, Put some hydrogen peroxide in a syringe with no needle, or an eye dropper. Squirt a few drops of hydrogen peroxide into the remaining algae. A few drops here, a few drops there, a bit more over there... Maximum H2O2 should be 3 ml /10 gallons. Wait half an hour then turn on the equipment. The concentrated H2O2 will kill the algae.
Alternative is to remove the Anubias and treat it outside the tank with a stronger dose of H2O2. Dip, let it sit in the water with H2O2, then give it a quick rinse in water removed from the tank, then return it to the tank. This is fine for any plant you can remove from the tank.
Beyond that, correct whatever conditions allowed the algae to grow in the first place.