We did the crossing in July of this year, doing so in two legs. The first leg we departed from Wales in Alaska and paddled west to meet a Russian team that departed the same day from Nouken in Russia. The 8 of us then paddled together to Inalik on Little Diomede Island (US) where we had prepositioned food, clothes, shelter and a support team.
Leg two was or paddling from Inalik north and west around Big Diomede Island to land near Uelen in Russia. We chose this approach for many reasons. You may wish to do the paddle after the ice breakup, paddling short distances and camping on the ice, the latter approach is potentially safer then the former.
The Bering Straits are notorious for bad conditions. Also, conditions can change really really fast, too.
On kayaks...I attached a link from Ray Jardine's site on building a double for the ice and abuse.
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