Learn Italian: Vidi la dea mentre beveva il vino. - I saw the goddess while she was drinking wine.

No, vedere (to see) is an irregular verb in the passato remoto. Instead of a regular ending, it changes its root to vid- and takes the irregular ending -i, making it vidi (I saw). You will also see this irregular root in the he/she form (vide) and the they form (videro).

Why does this sentence mix the passato remoto (vidi) with the imperfetto (beveva)?

This is a classic example of intersecting past actions. We use the imperfetto (beveva) for the ongoing background action (she was drinking). We use the passato remoto (vidi) for the sudden, single event (I saw) that happened while the background action was going on.

Is the word mentre (while) the reason we use the imperfetto here?

Exactly! Because mentre means while, it naturally sets up an action that is in progress. When you are telling a story in the past, mentre is a massive hint that the verb following it needs to be in the imperfetto (like beveva).

Why is there a definite article in il vino if the English translation is just drinking wine?