As outlined in Article 2 of the Constitution, the presidency is the collective administration of the chief executive of the United States. "Presidency" can refer to the term of office or the composition of the executive branch.

Elbridge Gerry supported the national executive being chosen by the state governors and it illustrates his growing concern about a powerful national government.  Gerry felt the new constitution would have to safeguard the righs of states and individuals.  He would later refuse to sign the new constitution.
Annotated by bacraig on December 05, 2014
In this passage, electors would vote for two people for president.  This proved to be problematic in the 1796 and 1800 elections, and it would need another amendment to the constitution to fix it (the Twelve Amendment).

In 1796, John Adams got the majority of the votes, but his party, the Federalists, also voted for a number of other people, which helped lead to Thomas Jefferson getting the second highest number.  Jefferson was from a different party, but yet, he became vice president.

In an effort to unify party voting, electors were encourage to vote the same party in 1800.  Voters expected Thomas Jefferson to be president and Aaron Burr as vice president, but in the end, the election was a tie.  It went to the House of Representatives, and Jefferson won the election.  When in office, Jefferson isolated Burr from any political discussion or policy decisions.
Annotated by bacraig on November 14, 2014
The residency rule was an attempt to block pro-British men who fled the country during the Revolutionary War.  Also, there were Europeans living in the country, like Baron Frederick von Steuben, who came to help the American military during the war.  However, if this one passed, some prominent delegates like Alexander Hamilton would not be eligible to be president right away.
Annotated by bacraig on November 13, 2014
Delegates felt that the president had to be a natural-born citizen or a citizen at the time of the constitution's adoption.

One possible reason to create this requirement was to stop any rumors that delegates were plotting to make a European monarch president of the U.S.  It was not unusual in Europe to have a foreign ruler, and one of these names being considered was Prince Henry of Prussia.

Another reason could be that delegates wanted to avoid a foreigner becoming a member of the cabinet or worse, commander-in-chief.  John Jay writes to George Washington on July 25, 1787: "Permit me to hint, whether it would not be wise & seasonable to provide a strong check to the admission of Foreigners into the administration of our national Government, and to declare expressly that the Command in chief of the American army shall not be given to, nor devolved on, any but a natural born Citizen."  Washington thanked Jay on September 2, 1787.

For the Prince Henry connection, see Richard Krauel. "Prince Henry of Prussia and the Regency of the United States, 1786". The American Historical Review, Vol. 17(October, 1911): 44-51.

George Washington's letter can be found in The Papers of George Washington Digital Edition, ed. Theodore J. Crackel. Charlottesville: University of Virginia Press, Rotunda, 2008. Confederation Series (1 January 1784–23 September 1788), Volume 5 (1 February 1787–31 December 1787): 271-272.
Annotated by bacraig on November 12, 2014
The Committee of Detail recommended that the president be at least 35 years old.  Delegates figured age brings maturity.

John Jay articulates the second reason in Federalist No. 64: "By excluding men under thirty-five from the first office, and those under thirty from the second, it confines the electors to men of whom the people have had time to form a judgment, and with respect to whom they will not be liable to be deceived by those brilliant appearances of genius and patriotism, which, like transient meteors, sometimes mislead as well as dazzle."
Annotated by bacraig on November 11, 2014
Elbridge Gerry was the first delegate to ask for qualifications for president.>

Up to this point, qualifications were discussed only in the context of the legislature, because they would be elected by the people.  In the beginning of the convention, delegates favored the legislature choosing the president among its own membership, thus qualifications were redundant.  However, at this point of the convention, it was more likely that citizens would choose the president, so qualifications were needed.
Annotated by bacraig on November 10, 2014
Along with many delegates at the convention, Charles Pinckney's opposition to the motion for executive appointment of judges was most likely rooted in his colonial experience of King George III and the royal governors.  Both the king and the governors used the appointment power as a patronage system and as a way to influence the legislature.  
Annotated by bacraig on November 06, 2014
By opposing James Wilson's motion for a single executive, this was the first time Edmund Randolph opposed the majority of his own delegation and the convention.  Randolph's biographer, John J. Reardon, argues, "There was no mistaking his mood.  He was incensed by the stubbornness of most of the delegates in refusing to recognize the dangers he felt were so obviously inherent in a single executive."
Annotated by bacraig on November 03, 2014
In this statement, Nathaniel Gorham argued against Madison's motion for the Senate to conduct a peace treaty on its own during war by saying Congress would have the power of the purse to cut off funding for a war.

In practice, it is very hard for Congress to unilaterally end a war.  Even during one of America's most unpopular war in Vietnam, it was not until June 1973 that Congress passed a provision to cut off funds for combat activities in Indochina.  President Nixon signed this into law months after the Paris Peace Accords were signed.
Annotated by bacraig on October 31, 2014
On August 6th, the Committee of Detail report said "The Senate of the U. S. shall have power to make treaties..." which gave a high-degree of power to the Senate regarding American foreign policy.

In reaction, Gouverneur Morris felt the House and president should be part of the treaty process by having treaties ratified by law.
Annotated by bacraig on October 29, 2014