Many lawsuits were directed at state judges before the election - I heard perhaps 300. The issue is this - the governor of the state, the secretary of state, and the state supreme court have no jurisdiction in the presidential election.
I do not know the details of how they stop at the state supreme court to move up to the Supremes, but that is not the main issue.
The US Constitution gives authority to the state legislature concerning voting rules and the election of the president. They send or refuse to send the electors to the big vote. The other side of this issue is the US House of Representatives. The US House of Representatives have to receive and verify the number of electors sent to them by the state legislatures. If a crisis means the House voting, then each delegation has one vote, so Rhode Island has one vote and California has one vote, etc.
So far, no one has been elected because the Electoral College has not met to vote (mid-December usually). There is no President Elect or Office of the President Elect. The new or old President takes over on Inauguration Day.
The Constitution does not have any deadlines except inauguration, Wednesday, January 20th, 2021.
