I've had a tank with a trickle down filter like that before. Honestly, I gave it a go for about 8 weeks, but ended up ripping it out and using an internal instead. I found it was difficult to keep a decent colony, seemed like too much air and not enough water, and the media went mouldy. I found that my tank was prone to ammonia and nitrIte spikes, and didn't seem to convert to NitrAte very well with the trickle filter.
I've kept P.signifier before, in fact, that was my first tank to have a heater malfunction in. They're nice little fish, but I was looking for something a little more striking. Surprisingly, blue eyes are not particularly easy to get here. No one really breeds them, so they're mostly wild caught at +$8/fish. It's also the wrong season for them, they seem to be round mostly in the spring.
I'm starting to like the look of Golden pencil fish, Nannostomas beckfordi. which sound easy-ish to breed, and seem quite attractive. Bonus points for being reasonably priced on my online LFS, but not so cheap that I won't be able to rehome them if/when they breed.
Do you have access to RO at all? You should be able to buy containers of it from your LFS if you are not able to have a plumb in one. Then you can safely remineralise and have perfect water all the time!
You would normally only replace ceramic media like biomax when it starts crumbling or breaking. When that happens, I like to slowly replace it 1/3 at a time, over several water changes, so that the drop in bacteria is not too rapid.
I believe your heaters are not keeping your tank up to temperature because of the way you have placed them. If you have them tucked away behind the slate, it's quite likely that they are only heating the immediate pocket of water they are in. Heaters generally work best when they are placed in the path of the filter outlet, so that they have good water circulation. Heaters have an internal thermostat, and if they are located out of flow they may switch off prematurely, as they only read the temperature of the water that they are in contact with. If you are not keen on visible heaters, an inline heater and an external canister filter may be more your style.
Unfortunately, I can't recommend any fish be kept in that size tank. It's just too small to keep stable, especially for a beginner. Shrimp or snails are really the only suitable inhabitants. You can have a very nice planted tank with shrimp though, they look fantastic against dark green plants like java moss. Shrimp are very easy to keep, very undemanding.
Worms may have appeared through excess un-eaten food, but this was not overfeeding, it was Pedro having a decreased appetite through cold water and ammonia/nitrIte poisoning. In cases like this, the best thing to do is continue the same feeding schedule (twice daily), but offer a reduced amount. A wider variety of food is also good, consider brine shrimp, artemisia and mysis in addition to bloodworm, and blanched peas de-shelled on for some veg.
As far as amount of food, a good rule of thumb is as much as they will take in 5 mins. It's been about 5 years since I last kept fighters, but I recall it being about a quarter of a block of food. Someone who's kept them more recently would be better placed to answer that.
NItrAtes are not your problem at 10ppm, rather, it's your NitrItes that are at 0.25ppm. In fishkeeping, in a fully cycled tank, Ammonia and NitrIte should always be 0, and NitrAte should be less than 40, though less than 20 is best.
You reduce NitrIte through regular water changes. If you still had fish in the tank, I would suggest two 25% changes on the first day, followed by one 25% each day until there is no nitrIte detected. I would also recommend considering Seachem Prime as your water treatment of choice, as it can be used at a higher dose to protect fish from the damage caused by NitrIte. More information is here.
As you do not have a fish in there, there is no need to remove the NitrIte until you restock, as the filter bacteria will slowly convert it to NitrAte.
One thing to consider is if you are not planning on immediately stocking the tank, you will need to continue to feed it to prevent your good bacteria from starving and dying. To do this, add the same amount of fish food to the tank that you normally would. It will break down and feed the bacteria.
Since it appears that LHG is not coming back anytime soon, should we consider having a section in the articles area for common diseases? I know we have a couple on things like Ich, but it would be good to have them grouped somewhere so they are easy to find, and we can link to them without sending people off site or having to type giant explanations.
I'd be happy to research and contribute to them also.
Re: A thought/concern about the fry I donate to my LFS
If the store is reputable, they may have them out the back in quarantine for a while also. I kink the store I used to sell to kept new arrivals off of the main grid for a week and on a tank with a steriliser until they were sure they were clean.
To me, this looks like saprolegnia, which is a kind fungus that exploits sub-optimal water conditions and weakened immune systems. It often attacks the gills of the fish, who then dies from haemodilution. It could also be columnaris, which is a similar looking bacteria. Saprolegnia tends to spread more rapidly in cool water conditions, so the low temperature of your tank may have contributed to this.
When you say that you change the biomax in your filter, how often are you doing this and is it all at once? If biomax are the little ceramic noodles, you should never need to change these, at least, not for years. They just need a swish out in some old tank water to remove any accumulated mulm. The noodles are where most of the good bacteria in your filter live, and it is very important that they are kept safe.
Your water results are showing a spike in nitrIte, which is poisonous to fish. This normally only happens if you have killed off some of the filter colony, or if you suddenly increase the bio load of the tank.
Fish, especially small fish, really need to be fed at least daily, preferably 2 or more times per day. Pedro looks quite thin in his postmortem photo, though not too bad in his swimming one.
I think that the series of events may have been as follows: Filter die off, leading to increased ammonia levels, combined with low temperature causes appetite supression and lethargy in Pedro. Reduced feeding schedule further combined with nitrIte spike further weakens fish, who then develops saprolegnia infection and dies.
In addition to all this bad news, your tank is also, unfortunately, too small to keep fish in for the long term. We recommend a minimum of 45L for a single fighting fish (more than twice the size of this tank). In small tanks, it is very difficult to maintain stable water chemistry, and things can go downhill very quickly, as you have experienced. You could consider keeping some cherry shrimp, as they have a tiny bio load.