inigo: As I understand it you have an extablished tank which has been moved. Plus you have been running it for a few weeks and nitrAtes went from 20 to 80ppm? You feed once per day, have an older box filter with no media. Also a CC substrate. there is red and dark algae algae in the box filter. You have also conducted a water change using ro/di water to no effect.
First of all the red algae is probably not corraline and therefore the result of phosphates. A normal thing for newly (re) established tanks. As the phosphates are consumed the red should be replaced with green algae.
Because you have not mentioned an algae problem, I presume you do not have one.
First I don't trust the nitrate test kits. that is from recent experience. With your livestock being ok and only feeding once per day, it could very be you have much lower nitrates. The test kit in my case was too sensitive. 20-30 PPM were reading 160+. But 0.0 was still reading 0.0.
Although upgrading your filtration may help, adding plant life either macros or marine plants someplace in your system will reduce nitrates. This will be true regardless of the filtration or water used. Additionally, the plants will consume phosphates, heavy ions, carbon dioxide, and generally balance out and stabilize the system. If you don't have any macros or plant life I recommed you add some.
Finally, as long as you livestock is doing fine, you may just want to wait and let things settle out.