Gotta love the ever-shifting word stress in Russian: one pronunciation mistake may make a sentence grammatically incorrect (or, still worse, change its meaning altogether)! With the stress on the last syllable (as in the recording), директора means "directors" (plural nominative). In the singular genitive and accusative forms the word's second syllable is stressed (диРЕКтора).
Contrary to what it may look like, the noun "имя" is not a masculine/feminine first-declension noun, but a heteroclitic neuter noun. You are correct in assuming that the verb "помнить" takes a direct object in the accusative; it's just that the noun's singular accusative form matches its singular nominative form.
Here's the complete declension table for "имя": Wiktionary/имя
Most nouns ending on "-мя" fall into the same category: бремя, время, вымя, знамя, имя, пламя, племя, семя, стремя, темя. They all have somewhat peculiar case endings.
It is the Genitive singular.
In Slavic languages, the direct object of a verb may switch to the Genitive when the verb is negated. It is not generally the case in modern Russian but was quite widespread as late as in the late 19th century:
- Я знаю имя → Я не знаю имя / имени.
- Я вижу папу → Я не вижу папу/ папы.
(of course, for animate masculine nouns like брат or мальчик, and all animate nouns in the plural, these forms are identical)
Today it depends on a number of factors such as the desired stylistic effect, the verb itself, stronger or more abstract negation etc. In the 19th century you just switched to the Genitive every time:).