Do (or did) any Protestant denominations believe in apostolic succession? If so, who do they consider the most recent valid Pope?
Yes a number of churches other than the Roman Catholic Church believe in the Apostolic succession. A good example is the Church of England, which also follows the commonly accepted definition of Protestant (though it considers itself somewhat different from most other Protestant churches, largely because of the belief in Apostolic Succession). Specifically they believe that the leaders of the Church of England also follow the Apostolic Succession (there is not necessarily only one Apostolic Succession - the Apostles appointed many leaders, and they in turn appointed many leaders etc.)
The situation is that the Anglican church does not deny the validity of the Pope's Apostolic Succession, or of his leadership of the Roman Catholic Church. It does (and more importantly, did in 1534) deny his status as a wordwide leader of all Christian churches. This is in effect the theological difference that led to the English Reformation and the formation of the independent Church of England, and eventually the worldwide Anglican Communion.
The difference of opinion in 1534 was that the Roman Church believed that all (Western) churches should owe their (earthly) allegiance to the worldwide church, which happened to have its seat in Rome. The founders of the Church of England believed that a national church should be independent. (I'm leaving out a lot of things that aren't relevant to the immediate question).
Therefore: the last Pope that Anglicans recognize as the leader of the Church of England would be the one immediately before the Act of Supremacy came into effect - Pope Paul III. However they would not deny the validity of all subsequent Popes, up to and including the current one, as leaders of the Church of Rome.