Question 1

Is Hamas a government or a terrorist organization?

Accepted Answer

Yes to both.  Hamas:
is an Islamist party that won election in Gaza in 2006, but then violently overthrew its coalition partner in 2007.  
Elections have not been held since. 
Freedom House rates Hamas' governance of Gaza as "Not Free", its lowest rating.
now rules Gaza as a military government.
is a "foreign-terrorist organization" as designated in 1997 by the US State Department, and dozens of countries since then.
opposes the Israel-PLO Letters of Mutual Recognition.
opposes the Oslo Accords, which:
established a peace process for the Israeli-Palestinian conflict, and
renounced  "the use of terrorism and other acts of violence."
is an Arabic acronym for "Islamic Resistance Movement".

The Council of Foreign Relations states in "What Is Hamas?":
The Palestinian militant group has struggled to govern Gaza and remains committed to violently resisting Israel.  Its surprise attack in Israel in 2023 threatens a wider conflagration in the Middle East.

For more information, see:
Wikipedia, Governance of the Gaza Strip 
Associated Press, What is Hamas? 
Foundation for Defense of Democracies. What Hamas Believes

Question 2

How does Hamas treat Palestinians?

Accepted Answer

In a CNN interview with the son of the Hamas founder, Mosab Yousef states;
Hamas does not care about the lives of Palestinians, Israelis, Americans, or their own lives.

The Foundation for Defense of Democracies writes in 10 Things You Need to Know about Hamas:
How does Hamas treat Palestinians?
Hamas diverts international assistance to build its terrorist infrastructure. This impedes economic development and contributes to the humanitarian and environmental crisis in Gaza. The group restricts the rights of women and Christians and treats homosexuality as a crime.

Hamas uses civilians as human shields, hiding terror tunnels, weapons factories, and operations centers under schools and in densely populated civilian areas. The United Nations (UN) has called on Hamas to “cease immediately” these violations of international law.

Question 3

What is the theology of Hamas?

Accepted Answer

Georgetown University Professor Ed Husain writes in The Theology of Hamas:
Hamas’s theology justifies its terrorism. “The so-called peace process is futile,” Hamas leader Khaled Mashal said in 2015, “There is no peace. Only the path of jihad, sacrifice and blood.”
The following sources present Hamas' organizing principles in its own language.
Hamas Charter
Wikiwand
Doctrine of Hamas
Wilson Center, 2023-10-20
(Congressionally-chartered research organization)
Text of the Hamas Charter
Yale Law School, 2023-08-18

Question 4

What do Hamas officials say?

Accepted Answer

In their own words,
Ghazi Hamad, Hamas Political Bureau, states "We will do this again and again.  [The October 7 attack] ... "is just the first time and there will be a second, a third, and a fourth... We are proud to sacrifice as martyrs" on Lebanon TV, translated by MEMRI.
Ahmad Kahlot, Head of Kamaal Aduan Hospital (in Jabiliya, Gaza) describes how Hamas brigade members double as hospital staff, Jerusalem Post (video).
Sam Abu Zuhri, Head of The Hamas Political Department Abroad, "We need violent acts against American and British interests everywhere", on Al-Aksa TV (Hamas-Gaza),  translated by MEMRI.
For more, see video compilation of Hamas officials on MEMRI.

Co-Founder's Final Message

Who Is Hamas?

Is Hamas a government or a terrorist organization?

How does Hamas treat Palestinians?

What is the theology of Hamas?

What do Hamas officials say?