When you say that you put a layer of old paper boxes against weeds, does that mean you put the broken down boxes first, and the put the compost on the top? If so, were the seedlings able to sink their roots through the paper boxes and go deeper into the ground?
I just put them dry, not broken, and overlapping between the boxes such that they cover the surface fully. And yes, the roots can go down, but weeds cannot go past that easily. The theory is yhe upwards growth is weak but downwards growth of the roots are much stronger and they can puncture a wet paper box.
Cover crops: clover, buckwheat and winter rye. Cut before seeding and lay them flat over the surface.
Just to clear up what sounds like a simple misunderstanding, because you mentioned being in Europe: in the US (can’t speak for other English speaking countries) to “break down” box is to flatten it, not break it like to destroy or tear apart. As in “help me break down these boxes for recycling”.
Forgive me if you got that, it just sounded like you were talking past each other.
I could see this being the reason your carrots didn't grow well though. If the carrots tap root struggles a bit through the cardboard it could mess up devlopment. I think this is why they say not to transplant carrots. The tap root bottoms out quickly and struggles to recover.
Cardboard doesn’t last long when it’s wet, but long enough to smoother the plants beneath it. There’s something about it that attracts the fungi that break down wood fiber. And the continuous surface allows it to spread quickly.
Also, what cover crops did you introduce?