The Kazakh SSR declared independence from the USSR later than the russian declaration meaning that for a few days the entire USSR was Kazakhstan. Is there any reason, except common sense, for which the Russia and not Kazakhstan took the position of the USSR in the UN and its nuclear weapons?

It's true that Kazakhstan was the last Soviet Socialist Republic to declare independence, but it's not really true that the Kazakh SSR was the USSR for a few days. Mikhail Gorbachev was still in Moscow working as the Soviet President, in charge of a Soviet government, to the end of the month. I go into further details of what happened in the few days prior to the Kazakh declaration of independence in an answer I wrote here.

As for why the Russian Federation gained the UN Security Council seat (and control of the Soviet nuclear arsenal, as well as responsibility for Soviet foreign debt), I go into that situation a bit here. In short, it was part of the agreement negotiated between 11 of the former Soviet Socialist Republics in the Alma Ata Protocol and signed on December 21, 1991.