The lead actor needn't memorize the entire script by tomorrow; we are only rehearsing Act I.

When need is used as a modal auxiliary (often in the negative), it does not take an "s" in the third-person singular and is followed by a bare infinitive (memorize). "Doesn't need" would require the full infinitive (to memorize).

As for the understudy, she dare not question my artistic vision during the performance!

Dare can act as a modal auxiliary. In the negative third-person singular present, it is dare not (or daren't). It does not take an "s" (dares not is incorrect here) and takes a bare infinitive.

Furthermore, the stage manager ought to have warned me about the wobbly chandelier before I stood directly beneath it!

To express a past unfulfilled obligation, we use ought to have + past participle (warned). "Ought" always requires the particle "to" in standard English.