I'll answer the first of your questions.

Quote Originally Posted by Ulamah View Post
...но я не помню много слов.
but I dont remember many words.

but here
слова короткие и грамматика простая.

They are both plural? слов and слова?
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).

Quote Originally Posted by Ulamah View Post
...I think I will stick with
Я не помню многие слова for the first sentence.
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.