Note that 1 & 3 are true, but for completely different reasons; the UNSC has non-permanent members, and Kazakhstan was elected to one of these for 2017-2018.
The claim 2 seems specious. The UN[SC] seat of the USSR was renamed to Russia, and so for a while the Kazakhstan SSR/Republic was formally represented at the UN via Russia, while their application for a separate seat was being processed. As the Kazakhstan representative (Akmaral Arystanbekova) recalls:
On 21 December 1991, at the meeting of heads of sovereign States in Alma-Ata (Kazakhstan) where the Commonwealth of Independent States (CIS) was created, the decision was made for the Russian Federation to succeed the United Nations membership of the USSR and for the other former Soviet republics to accede to the United Nations.
On 23 December 1991 my first meeting took place with Yuli Vorontsov, the Permanent Representative of the USSR to the United Nations. I knew him well, as in autumn 1990 I had participated in the forty-fifth session of the United Nations General Assembly as Minister of Foreign Affairs of the Kazakh Soviet Socialist Republic (SSR) and a member of the Soviet delegation. At the meeting I gave the Ambassador a copy of a letter from President Nazarbaev to Boutros Boutros-Ghali, the newly elected United Nations Secretary-General, in which he congratulated Mr. Boutros-Ghali and noted that he had sent Minister of Foreign Affairs Akmaral Arystanbekova as his representative to the Permanent Mission of the USSR to interact with the United Nations Secretary-General. We agreed that my duties would be determined by the interests of the Republic of Kazakhstan. The Ambassador stressed that the Permanent Mission would provide every possible assistance for the implementation of my main task: preparing the way for Kazakhstan's accession to the United Nations.
On the next day, 24 December 1991, the Ambassador assembled the diplomatic personnel of the Mission and announced that he had presented a letter from President Boris Yeltsin to the then-Secretary-General, Javier Pérez de Cuéllar, stating that the United Nations membership of the USSR, including its membership in the Security Council, would be succeeded to by the Russian Federation, with the support of the CIS countries. The Permanent Mission of the USSR to the United Nations would be renamed the Permanent Mission of the Russian Federation to the United Nations. Thus I became a Senior Counsellor at the Permanent Mission of the Russian Federation to the United Nations, representing the Republic of Kazakhstan. Several years later Perezi Kamunanwire, Uganda’s Ambassador to the United Nations, told me that the Ambassadors of the African States had appealed to the United States Permanent Representative to give the Soviet Union’s place on the Security Council to a State from Africa, which is not represented among the Security Council’s permanent members.
On that same day, 24 December 1991, I sent to the President of Kazakhstan information about the documents required to accede to the United Nations, which I had received at the United Nations Secretariat, and asked for an application for admission to be sent by 31 December 1991.
On 31 December I sent to the United Nations Secretary-General’s Office a faxed letter of the President of the Republic of Kazakhstan and an official application for admission to the United Nations, signed by President Nursultan Nazarbaev. [...]
On 21 January 1992, the Committee on Admission of New Members unanimously agreed at its meeting to recommend that the Security Council admit the Republic of Kazakhstan to membership in the United Nations. [...]
On the historic day of 2 March 1992, the United Nations General Assembly, at the eighty-second plenary session of its forty-sixth session, unanimously admitted the Republic of Kazakhstan to the United Nations.
So, roughly speaking, between 24 December 1991 and 2 March 1992,
Kazakhstan was still being "represented" by... Russia at the UN.
I put this in scare quotes because de facto Kazakhstan had their own representative with whom to conduct talks etc., but she/Kazakhstan didn't have a UN seat (let alone a UNSC one) for formal things like votes and so on.
