There were 7 presidents before George Washington under the Articles of Confederation who served one year terms each from 1782-1788 (actually it was started in November 1781 and ended in April of 1789 technically when George Washington took office for those who split hairs). They had nowhere near the power of the US president under the US Constitution and we don't even tend to count them as true US presidents, even thought they technically were.
These presidents were John Hanson (1782) Elias Boudinot (1783), Thomas Mifflin (1784), Richard Henry Lee (1785), Nathan Gorman (1786), Arthur St. Clair (1787), and Cyrus Griffin (1788).
The current requirements to be US President under the US Constitution are:
1. Be 35 years old or older on the day you will be sworn into office (usually Jan 20th of the year following the election)
2. Be a US citizen by the time of being sworn in.
3. Be born in the USA. Being born in a US territory doesn't count. However if one is born in a US territory and it becomes a state by the time that candidate is sworn in, that would almost certainly be considered acceptable to meet this requirement (since the first several presidents were not born after July 4th, 1776 and couldn't be since no one born in 1776 could have been at least 35 years old in 1789, the first swearing in for a US president under the US constitution).
If statehood was ratified retroactively (as was done for Ohio when a mistake was made where everyone thought it was officially ratified for over a century but it wasn't since not all papers were properly signed), that is acceptable if that candidate was born after the retroactive date of the ratification (however retroactive acts are not allowed by the US constitution, but that wouldn't stop Congress as long as a court never had a reason to review it. Even if a court reviewed a retroactive act, courts have often ignored that retroactive acts of Congress should not be allowed.)
4. Either win the electoral vote as counted and evaluated by the new House of Representatives (Usually on Jan 6th of the following year) or if no one wins 50% + 1 of the votes then it's decided by the House where each state gets one vote. Any of this may be contested in the Senate if one Representative and one Senator files a protest. In a protested or a corrupted election (which happened in Tilden vs. Hayes in 1876) then the House may follow any procedure it deems fit. The precedent set in 1876 was for an electoral commission appointed by the House whom would decide which candidate won which electoral votes. However in 1876 this turned into a very partisan affair and the party in power gave it to their party's candidate by contradicting themselves a couple of times to make sure their candidate (Hayes) won all 8 contested states (all of which Hayes had to get to win while Tilden just needed any one of them to win)
Requirements to become US president under the US Constitution may only be changed by adding an Amendement to the US Constitution. This requires a 2/3 vote of Congress, 3/4 of the states approving, and the president to sign it. There have only been 27 amendments to the US constitution since its full ratification in 1788. The 27th amendment was ratified in 1992.
In the 2000 election, if it went to the House as it should have gone instead of the Supreme Court, Bush would have won in the House since it's one vote per state and Bush won more states. Also the fact that the new House of Representatives that decided were mostly Republicans helped. The Supreme Court saw that the Presidency would wind up going to Bush even in a contested or protested election in the House under the rules in every scenario. They wanted the matter settled for the sake of the stability and defense of the country. However, they decided to end the matter early by making an arguement that only when Gore tries to win a presidental election in 2000, then states rights are violated. It was decided by judges mostly picked by Republican presidents. Two of those judges (Scalia and Thomas) both were involved in the Bush campaign and didn't excuse themselves. However there is no higher court than the Supreme Court, so no one but the Supreme Court could have judged this unacceptable.
Ironically, if a recount of Florida was done in a way Bush's team wanted, Gore would have won. If a recount of Florida was done in a way Gore's team wanted, Bush would have won. Even when newpapers finally finished recount of their own months later, it was still unclear since it depended on which criteria ballots were acceptable. This didn't matter since any complete and thorough recount couldn't have been done by Jan 6th the real deadline and not December 13, the made up one announced to the world. Therefore the election still would have been in the House's hands and Bush would have won in all scenarios.
The Vice President would have been decided by the Senate where each senator gets one vote. Therefore since Lieberman was in the new Senate, he would have been allowed to vote. It was 50-50 between Republican and Democrats in the new Senate on Jan. 6, 2001, so it would have been a tie. That means that Gore would cast the deciding vote since he presided over the new Senate until Jan 20th 2001. He could have then made himself Vice President for a 3rd term under Bush if he so wished and there are no term limits on vice presidents. They may have also opted for Lieberman to be Vice President under Bush.
In this case, who was vice president did matter, since it was 50-50 in the Senate. When Gore presided over the New Senate for 2 weeks, it was a Democratic majority and afterwards it was a Republican majority when Chaney became Vice President. Therefore legislation was seriously effected depending on who was vice president in this case and the Supreme Court unfairly fixed it early and made the Republicans regain power in the Senate after Gore left office. However Senator Jeffords of Vermont changed from Republican to independent several months later and aligned himself with the Democrats in voting to give the majority back to the Democrats. However Chaney could still cast a tie-breaking vote even after Jeffords switched parties.
If the mess would have lasted until Jan 20th 2001 (and that was possible since there is no time limit for the decision of the House), Clinton and Gore still had to step down on January 20th 2001. With no clear president or vice president, the 3rd in the chain of succession would be next selected, which is the Speaker of the House as of Jan 6th 2001, which was Dennis Hastert. However it would only be until a new president was sworn in. Hastert would have had to resign his seat in the House even for this temporary appointment and could only get back in by being elected (almost certainly at least 2 years later). However he's not required to take the post if he doesn't want it and probably wouldn't want to give up being speaker to be president for a day or two. Therefore the 4th in line would be asked. That was 98 year old President Pro-Tempre of the Senate Strom Thurmond. That was his last term as a senator anyway and would have certainly wanted the position for a few days at the cost of his senate seat. However he was also very senile by that point and having him in charge of our nuclear arsenal is a scary thought.