Learn Dutch: Mij wordt in het casino een cocktail aangeboden. - I am offered a cocktail in the casino.

In English, you can make the indirect object the subject of a passive sentence, like "I am offered". Dutch does not allow this. The thing being offered (the direct object, een cocktail) becomes the subject. The person receiving it remains an indirect object, which is why we use Mij (to me) instead of Ik.

If Mij is at the beginning, why doesn't the verb conjugate to match it?

The verb always agrees with the grammatical subject. Since Mij is an object, it cannot control the verb. The actual subject of this passive sentence is een cocktail (a third-person singular noun), so the auxiliary verb must be wordt.

Could we use Aan mij instead of just Mij here?

Yes, you can! Because Mij functions as the indirect object, you could add the preposition and say Aan mij wordt in het casino een cocktail aangeboden. Dropping aan and just putting Mij in the first position is perfectly correct and very common in Dutch.

The past participle at the end is aangeboden. What is the infinitive, and why is the ge in the middle?