In what ways are Lucifer, Minos and Beatrice tortures extreme version of their lives and sin on earth? what connection does dante see between the punishment and the sin
First, Beatrice is in Heaven, not hell, and thus has no punishment. Her role in the Inferno is as Dante's love interest. Because Dante loved Beatrice in real life, he places her next to the Virgin Mary and St Lucia in Heaven, thus portraying her both as holy and as the symbol of beauty and love. Beatrice asks Virgil to guide Dante through hell.
Minos is actually a pagan god, and thus never lived as human. Dante writes him in the Second Circle of Hell as the one who places all sinners in the correct level of hell (he wraps his tail around them to indicate which level-- quite clever!). His presence in hell is significant because it proves that Dante promotes Christianity-- otherwise, he would not put a god in hell!
Lucifer is of course at the very last level of hell (the Ninth Circle), since he was the ultimate sinner. As punishment for defying God, Satan is stuck for eternity in a frozen lake. Thus Dante is mocking Lucifer's previous desire for freedom. Plus, the lake stays frozen by the wind from Lucifer's giant wings, which means that he is always causing his own punishment. Also, Dante writes Lucifer as a giant with 3 heads, each of which perpetually eats the 3 greatest traitors in history: Brutus, Cassius, and Judas Iscariot. Lucifer's three heads is a contrapasso on the Christian trinity.