The International Criminal Court's jurisdiction
Is it true that the International Criminal Court can only bring a
state to trial if that state accepts the court’s jurisdiction over
them?
The International Criminal Court (ICC) can't bring a state to trial. It only has jurisdiction over individuals accused of the war crimes and crimes against humanity over which the ICC has jurisdiction. As explained at the Wikipedia link above:
The International Criminal Court (ICC or ICCt) is an intergovernmental
organization and international tribunal seated in The Hague,
Netherlands. It is the first and only permanent international court
with jurisdiction to prosecute individuals for the international
crimes of genocide, crimes against humanity, war crimes and the crime
of aggression. The ICC is distinct from the International Court of
Justice, an organ of the United Nations that hears disputes between
states. . . .
As of February 2024, 124 states are parties to the Statute of the
Court, including all the countries of South America, nearly all of
Europe, most of Oceania and roughly half of Africa. Burundi and the
Philippines were member states, but later withdrew effective 27
October 2017 and 17 March 2019, respectively. A further 31 countries
have signed but not ratified the Rome Statute. The law of treaties
obliges these states to refrain from "acts which would defeat the
object and purpose" of the treaty until they declare they do not
intend to become a party to the treaty. Four signatory
states—Israel in 2002, the United States on 6 May 2002, Sudan on 26
August 2008, and Russia on 30 November 2016 —have informed the UN
Secretary General that they no longer intend to become states parties
and, as such, have no legal obligations arising from their signature
of the Statute.
Forty-one additional states have neither signed nor acceded to the
Rome Statute. Some of them, including China and India, are critical of
the Court. Ukraine, a non-ratifying signatory, has accepted
the Court's jurisdiction for a period starting in 2013. . . .
For an individual to be prosecuted by the Court either territorial
jurisdiction or personal jurisdiction must exist. Therefore, an
individual can only be prosecuted if he or she has either (1)
committed a crime within the territorial jurisdiction of the Court or
(2) committed a crime while being a national of a state that is within
the territorial jurisdiction of the Court.
The ICC only has jurisdictions over individuals who commit crimes within the ICC's jurisdiction in states that have assented to the treaty or while they are citizens of states that have assented to the treaty.
The International Court of Justice's jurisdiction and authority
If this is the case, has the ICJ never been able to declare a state
leader a war criminal while that leader was in power, assuming that
the state leader declined to participate?
The International Court of Justice (ICJ) is a UN affiliated court that exists to adjudicate disputes between states and is not a criminal court. It does not have jurisdiction to hold any individual person civilly or criminally responsible for anything.
Historically, the ICJ has most often been utilized to resolve boundary disputes between countries with the mutual consent of the countries that are parties to the case. This is because (from the Wikipedia link above):
The International Court of Justice (ICJ; French: Cour internationale
de justice, CIJ) is the only international court that adjudicates
general disputes between nations, and gives advisory opinions on
international legal issues.
Thus, ICJ decisions aren't binding on member states. The fact that they are non-binding is largely a reflection of the fact that there is no means to enforce an ICJ decision on a member state.
It is also correct, as noted in the question, that Article 36 of the ICJ statute basically allows a member state to decide for itself under what circumstances it will be subject to the ICJ's jurisdiction.
How the United Nation's Works
Is there no clause that by choosing to be a part of the UN you also
have to follow the UN’s laws, or anything remotely similar?
United Nations (U.N.) members have to abide by U.N. procedural rules for how it conducts business, although the U.N.'s enforcement powers, if, for example, a U.N. member doesn't pay its dues, are very weak. As explained at the Wikipedia link above:
The UN has six principal operational organizations: the General
Assembly, the Security Council, the Economic and Social Council, the
International Court of Justice, the UN Secretariat, and the
Trusteeship Council, although the Trusteeship Council has been
inactive since 1994. The UN System includes a multitude of specialized
agencies, funds, and programmes, including the World Bank Group, the
World Health Organization, the World Food Programme, UNESCO, and
UNICEF. Additionally, non-governmental organizations may be granted
consultative status with the Economic and Social Council and other
agencies.
The core U.N. organization is basically a talk shop and a multi-lateral forum in which diplomats from member nations can interact with each other with relative ease. It facilitates diplomatic activity, but it is not, and is not intended to be, a global super-legislature with executive and legislative authority over member nations.
The U.N. Security Council can give its blessing to enforcement actions organized and funded by member states, but the existence of multiple permanent Security Council members with veto power has greatly limited its use in this manner. The United Nations does not have an army of its own. "Blue helmet" peace keepers operating under U.N. Security Council authorization are contributed by member states on a mission by mission basis, rather than being part of a U.N. standing army