Increasing sea ice is an effect of global warming?

That David Rose Daily Mail piece on increased Arctic ice may prove the opposite of what he claims. Far from demonstrating global cooling, it may be a side-effect of warming, plus deterioration of the ozone layer.

This article from Australia in 2011 (I'm asking for permission to reproduce in full) argues a causal chain, which I understand as follows:

1. The hole in the ozone layer above the South Pole is cooling the stratosphere
2. An effect of this local temperature change is to let air drop down and push out, strengthening the winds that circle the Antarctic
3. The boosted cyclonic winds push sea ice around and create gaps within the sheet, where the water is less turbulent and so freezes more easily
4. Meanwhile, the warmer air causes more rain and snow to fall on the sea*
5. This (less salty) precipitation is less dense than the warm saltwater rising from lower layers of the sea, so it sits on top and keeps the surface cooler than it otherwise would be
6. The lower, warmer water is largely trapped in its own layer, so less of it rises to melt sea ice

*I don't know whether this is because warmer air carries more water vapour and so has more to dump when it meets a front of colder air; or for some other reason.

I've asked the writer if the same thing could be happening in the Arctic.

Also, if the sea is warmer, then presumably there will be more evaporation and so more to fall on the central ice cap of the North Pole and the landmass of the Antarctic, potentially increasing the thickness of the ice at each place.

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