Good questions, thank you Macid.
Answering your last question first, my thermal conductivity calculation was for an illustrative 'toy model' that might approximate the true situation in some circumstances. The whole point is to get the magnitude of the heat flux for the process of thermal conduction -- which was found to be of order 1/100 W/m^2.
As you point out, this is negligible compared to the other thermal processes effecting the melting or freezing of the ice, which was what the toy model was intended to illustrate.
The background to my post above has been the common narrative on this forum that excess heat gets stored in the water column over the course of the melt season, together with the assertion that the refreeze cannot proceed to any great extent until that excess heat has first been removed by extraction to the atmosphere.
I've never been comfortable with that explanation because, given the situation of relatively calm water, I can't see how that trapped heat -- which I presumed to extend down by at least several meters and probably tens of meters or more, depending on whether it was sourced from ocean currents or direct sunlight -- can get to the surface to influence the refreeze.
So yes, as you point out, in calm conditions then the long-wave radiation to the cold sky should start the refreeze without noticing how much excess heat has been trapped below. Then, once the ice has formed and grown thick enough, there is no chance for wind and waves to start bringing the excess heat up to the surface. So the re-freeze continues and the trapped excess heat below becomes essentially irrelevant.
My suggestion above concerns the converse situation, where there is wind and waves. I'm suggesting that could mix some of the excess heat up to the surface to replace the heat lost to long-wave radiation and so to retard the start of the re-freeze.