If a US president becomes president "by accident" instead of by election, is this president eligible to run for two terms thereafter?

The Constitution explains this quite clearly.

If you serve more than 2 years of the other person's term, you can run for ONE additional term.

If you serve less than 2 years of the other person's term, you can run for TWO additional terms.

So the maximum length of time a person can be president is 10 years: 2 years of the dead guy's term, and 8 years of your own.

This is why LBJ could serve part of JFK's term, then be elected in his own right in 1964, then run for a second full term in 1968 (although he eventually dropped out).

if he became president with 2 years or less left on that term, he would be eligible to serve two terms of his own. Harry Truman served more than 3 years of Roosevelts 4th term and elected for a term of his own. He was eligible to run again in 1952 because the term limit amendment excluded him. He chose not to run again Lyndon Johnson served less than 2 years then a term of his own. He was eligible but chose not to run. Ford was defeated when he tried to run for a term of his own. Having served 3 years of Nixons term, he would not have been eligible for a second term of his own had he beat Carter.

Yes, but none has ever gotten that far.