Interesting questions Rox. A couple of general points (based on my understanding, and I'm happy to be contradicted) before trying to answer some of your specific questions. Sea ice, particularly first-year ice, contains pockets of salty brine. This initially remains liquid when the sea ice freezes, as it has a lower melting point, and some of it then drains out, leaving air pockets. Some remains as trapped pockets of brine, which melt at a lower temperature than the surrounding ice. This, as well as other factors, mean that there are weaker and stronger points within the ice, including potential routes for the water to escape from the surface to the sea. When melt water refreezing blocks these cracks/passages, it doesn't necessarily block them at the base of the floe. If surface melt has started before the ice is all at 0C, the water will refreeze at some point on its way through the floe.
The water in melt ponds can reach temperatures above zero.
Fresh water coming into contact with subzero salt water won't necessarily freeze on contact. It may manage to mix sufficiently before losing enough energy to its surroundings to change state.
Now to your questions:
1. Yes, probably in general, but only at the column where the water drains. The whole floe won't be the same temperature. It is also possible that "warm" (1 degree?) melt water is able to escape before refreezing through a larger crack produced by dynamic processes.
2. Usually, but not always, I think. From observation, sometimes you see melt ponds drain in areas where it is unlikely that the water has reached 0. The buoy temperature profiles could help with this.
3. Not sure, but I don't think so. See 2 and the point about losing energy to change state.
4. Yes, it must be thick and strong enough, but I think the depth implied by this varies a lot. Sometimes floes melt out very quickly after melt ponds drain, sometimes slowly.
5. Floes are massive, and they are constantly losing and gaining heat in different places, and heat is being transferred through them. The temperature won't be uniform when the pond drains. Either way, the energy needed for the change of state is much greater than the energy needed to raise the temperature a degree or two, so I don't think this has a huge impact on the subsequent rate of melt.
Overall, I think you're underestimating the dynamic nature of the process and the variation in conditions within avsing floe.