Can the Senate or any government official initiate peace talks or negotiations during a war without the President's consent? Or does this amount to treason? I read that there's this discussion among legal experts that every person appointed by the President works for him, so I am wondering if there are people outside of the Senate who may challenge the president, and challenge the president by trying to negotiate or talk for peace for a war that was declared by the President.
No, negotiation requires the authority of the United States. The Logan Act states:
Any citizen of the United States, wherever he may be, who, without authority of the United States, directly or indirectly commences or carries on any correspondence or intercourse with any foreign government or any officer or agent thereof, with intent to influence the measures or conduct of any foreign government or of any officer or agent thereof, in relation to any disputes or controversies with the United States, or to defeat the measures of the United States, shall be fined under this title or imprisoned not more than three years, or both.
This doesn't say explicitly that the authority of the president is required. That is stated in the constitution which vests the authority to negotiate treaties with the president. As Wikipedia states:
Article II, Section 2, Clause 2 of the United States Constitution includes the 'Treaty Clause,' which empowers the President of the United States to propose and chiefly negotiate agreements, which must be confirmed by the Senate, between the United States and other countries, which become treaties between the United States and other countries after the advice and consent of a supermajority of the United States Senate.
[The President] shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two-thirds of the Senators present concur ...
In United States v. Curtiss-Wright Export Corp., 299 U.S. 304 (1936), Justice Sutherland, writing for the Court, observed,
[T]he President alone has the power to speak or listen as a representative of the nation. He makes treaties with the advice and consent of the Senate, but he alone negotiates. Into the field of negotiation, the Senate cannot intrude; and Congress itself is powerless to invade it. As Marshall said in his great argument of March 7, 1800, in the House of Representatives, 'The President is the sole organ of the nation in its external relations, and its sole representative with foreign nations.'
Or does this amount to treason?
No, violating the Logan Act is a felony. Whether a specific act that violates the Logan Act is also another crime (like treason) depends on the specifics of the case.
I read that there's this discussion among legal experts that every person appointed by the President works for him, so I am wondering if there are people outside of the Senate who may challenge the president, and challenge the president by trying to negotiate or talk for peace for a war that was declared by the President.
In 1798, he went to Paris to negotiate peace with the French to settle the Quasi-War. On his return, he found he had been denounced by the anti-Jeffersonian Federalists, who had passed a statute informally known as the "Logan Act", which made it a crime for an individual citizen to interfere in a dispute between the United States and a foreign country.