Learn Finnish: Sankarin ei tarvitse paeta. - The hero does not need to flee.

Because it is used with the necessive verb structure ei tarvitse. Just like with täytyy and pitää, the person experiencing the necessity (or lack of it) must be in the genitive case.

Is ei tarvitse the standard way to say someone 'doesn't have to' do something?

Yes! To express that there is no obligation, Finnish uses ei tarvitse (does not need to) as the direct negative equivalent of täytyy (must).

If we wanted to use the 3rd-person imperative from this lesson to say 'Let the hero not flee!', how would that look?

It would be Sankari älköön paetko. Notice that the subject returns to the nominative case (sankari), the negative verb becomes älköön, and paeta takes the imperative connegative suffix -ko.

Why doesn't tarvitse have a personal ending here, like tarvitsee?

In the negative present tense, the main verb drops its personal ending and takes the connegative form. The negative word ei already tells us it's the 3rd person singular, so tarvita becomes tarvitse.