I'll answer the first of your questions.
Yes, "слова" (I've underlined the stressed vowels) are "words" (Nominative Plural)
"слов" is the Genitive Plural form.
Singular Plural Nominative слово слова Genitive слова слов
"много" (as well as the most of numerals) requires a word to be in the Genitive Plural (a lot OF words).
It's OK, and it even sounds better.
But the most natural way to say it in Russian is "Я не помню многих слов" (Genitive of negation).
Compare:
Я помню многие слова. - I remember many (of) words.
Я не помню многих слов. - I do not remember many (of) words.
Although, "Я не помню многие слова" is not wrong.