Learn Italian: Mentre affetta il tacchino, mi dà anche dello spumante? - While you slice the turkey, can you also give me some sparkling wine?

The partitive article (meaning "some") is formed by combining the preposition di with the definite article of the noun. Because spumante is a masculine noun that begins with an s followed by a consonant, it takes the definite article lo. Combining di + lo gives us dello. If the noun started with a regular consonant, like vino, you would use del (di + il).

Is it grammatically required to use dello here, or could I just say mi dà anche spumante?

You could technically drop it, but using the partitive article (dello, del, della, etc.) is much more natural in everyday spoken Italian when asking for an indefinite portion of a mass noun like a food or beverage. Dropping the article entirely to mean "some" is more common in negative sentences, broad general statements, or long lists.

The English translation says "While you slice," but aren't affetta and dà the he/she forms of the verbs?

Yes, grammatically they are third-person singular. The speaker is using the formal Lei to address the person politely—likely a waiter, caterer, or deli worker. If they were speaking informally to a friend, they would use the second-person tu forms: Mentre affetti il tacchino, mi dai anche...