The idea in my head right now is that at certain times Italians merge words together, when SPEAKING them. So in this instance: "La donn- ha una forchetta" (they don't pronounce the 'a' at the end of the word 'donna')
If it was 'the man has a fork', the merging would be slightly different. It would be: "l'uomo ha -na forchetta" (they merge ha and una - they won't pronounce the u at the beginning of 'una').
Is this correct?
Most languages tend to join words together to form a kind of melody, and the sentence is spoken as if it were a single word; in proper Italian that "merge" mostly happens when two of the same vowels are next to each other, as Italian doesn't have any word like the French fenêtre, i.e. there are no native double vowels. In most other cases the merge or the loss of parts of the word is indicated in written form too (truncation or elision). The case of una becoming 'na is common is some dialects (many in the Center and South), but it's not proper in Italian.
That particular type of merging led to forms like compound prepositions (del, al, nel, ...), clitics and demonstrative pronouns, but their original form isn't in use anymore. Truncation (e.g. amore becoming amor or uno becoming un) and elision (e.g. lo and la becoming l' and una becoming un') are phoenomena more easily recognizable.
The reason is the same, although the outcome is very different. The liaison is a case of enchaînement, the rule that when the following word starts with a vowel the words are concatenated: this in French usually leads to consonant endings being spoken as if they were part of the following word, while in Italian to vowel endings being overshadowed by the following word.
"E" is never pronounced "ee" in Italian, and I don't hear it in this recording either; perhaps it feels that way because the double consonant shortens the vowel a bit, but the English short "i" just doesn't exist in Italian (hence why many Italians pronunce "slip" as "sleep"). The only rule is the dictionary, and many regional accents like to open their vowels, so while it's forchétta (ay sound), you'll hear it as forchètta (eh sound) as well.
"Donna" isn't any less rude than "woman"; it used to mean "mistress" a few centuries ago (archaic and literary, says Treccani), and it's still used like that in some places (mostly as the feminine form of "don"), but in mainstream Italian it just means a mature female specimen of human, and I wouldn't address one as such, especially when she's holding a fork.