What would happen if no one ran for president?

[True Federalist (진정한 연방 주의자)]

As far as the Constitution is concerned, no one votes for President on Election Day, rather the State Legislatures have chosen to allow them to vote for Electors. If for some strange reason, all 538 Electors nominate different people, then it would be a wide open race in the Congress with the House having 538 choices for President and the Senate having 538 choices for Vice President.

[True Federalist (진정한 연방 주의자)]

In California, write-in votes are discarded unless a candidate registers as a write-in candidate.  Is it like that in other states?  Because if so, no write-in votes would be counted.

South Carolina does not allow for write-in votes for Presidential Electors.

[True Federalist (진정한 연방 주의자)]

What if the Senate were still able to pick the Vice President? Do they go under different rules?

Somewhat, yes.

If the House hasn't settled on a President yet by the time term starts, the VP-elect serves as Acting President. That question is settled IIRC.

Also if there is no President-elect or Vice President-elect chosen by the start of the term, the Speaker serves as Acting President, if eligible.