[Zapins]

The tank is a 54l/14gallon with an old 15w t8 18000K POwer-glo from when I quit the hobby a couple years ago.

18000K is not an appropriate freshwater bulb. Most of the light the bulb puts out is unusable by freshwater plants. You should buy one that is about 6500K. I think this is your main problem. Also, 15w over a 14g tank is low light, perhaps too low to grow anything no matter what fertilizers you put in it. Can you add another 15w bulb so you have 2 on your tank? Even for low light it is on the ultra low side, moss and ferns might work.

Soil is two fists of boiled peat covered by 8 litres of some burned clay substrate that I don't know the name of.

An unusual substrate, it might work, but it is not something many people use. It will likely make the water quite soft.

mixture of 25ml of micro+ in 500ml of water

micro+ is Fe 20g/l, Mn 10g/l, Cu 0.52g/l, Zn 2.9g/l, B 4g/l, Mo 0.75g/l

From this I have dosed 5ml of KNO3, Kh2Po4 and K2SO4 on day 1,5 and 7

Also dosed 5ml of Micro+ on day 5 and 7

I'm not familiar with Micro+, or is it the same thing as intermag mikro plus?

No technology has been used in the tank, no filter or heater. Could this maybe be the issue? Lack of water circulation?

This is certainly a problem. Water circulation is necessary for proper plant growth and it helps tremendously. I don't think it is the cause of your current problem (you need more light), but it will probably cause issues down the road if your don't get a small powerhead or device to move the water around. You don't want to have something splashing in the tank like a bubbler, but you do want water to circulate and prevent deadspots from appearing.

The guy in the store said that the soil would create circulation through bacterial activity that would create warm water in the soil which would rise. It is also meant to work as the filter at that point, but it is maybe not very active now without fish and daily feeding?

Bacteria will create a small amount of circulation but I'd argue not enough to solve the problem. You really do need a small pump or something in the tank to move the water around and prevent it from going stagnant.

Lights are on 9-13 and 15-20

Why the break?

Green color fades on some plants on the ends of the leafs or whole leaf.
Brown/red spots on some plants and in general the plants don't look as sturdy.

I'll wait to see the photos before giving a more concrete answer but this description coupled with the [lack of] lights is most likely the reason the plants are dying.

 

[Zapins]

Actually, the Hagen 18,000k DOES support FW plant growth. If you look up the bulb specs, it has a nice red peak.

Perhaps you meant blue? This bulb has almost 0 red in it. From the website:

18,000K lamp illuminates aquariums with super bright, bluish white light

Spectrum for Power-Glo Fluorescent Tubes

A really nice summary of light can be found here:
http://www.aquaticplantcentral.com/forumapc/lighting/38014-lighting-spectrum-photosythesis.html

These are usable wavelengths of light:

So yes technically the bulb has some usable light in it, but it is not ideal, you are missing the red side of the spectrum completely. T8 bulbs are also not particularly intense, and reflectors for them aren't particularly effective due to the thicker bulb diameter.

As lauraleellbp already mentioned the bulbs can be used to grow some species of plants. But they probably can't grow the ones you have listed since they require higher light.

 

[Zapins]

The lighting rule breaks down for tiny and huge tanks. 40 watts over a 5g tank isn't unheard of depending on the type of bulb used. There is a minimum threshold that plants have for light. I agree that 2x15w fluorescents will probably do the trick.