Are Roman Catholics required to believe that the Cardinals choice of a new Pope is inspired by the Holy Spirit?
If so, how do we explain "bad" popes?
One of the principles of denominations which give importance to the sacraments is that the unworthiness of the priest does not invalidate the sacrament. Even if your priest has himself sinned, that does not prevent the grace of God acting through that priest in the sacraments. Though I can't find a clear statement about it I would expect a similar principle to apply to popes - that God can use an imperfect pope to perfectly guide his church.
The Holy Spirit should be invoked and should guide the election of the Pope.
From Bl. John Paul II's Apostolic Constitution:
That said, there remains the possibility that the cardinals will fail to discern the will of the Spirit. However, that can no more be determined by the faithful than it can be by the cardinals themselves. So yes, we must believe that the right person has been chosen.
DJClayworth is correct that, even if the wrong person is elected, his ministry is not invalidated. God will find a way of making everything good.