Short answers, pulled from the story.
The official Latin term was rector provinciae, used in Roman legal language regardless of the specific title the governor held. Other titles such as proconsul, propraetor, and praeses reflected the province's status and the governor's rank.
Yes. The governor held the sole right to impose capital punishment in his province, and capital cases were normally tried directly before him. Appealing his verdict required traveling to Rome to argue before the praetor urbanus or the Emperor, an expensive and rarely successful process.
During the Republic, the Senate appointed provincial governors as promagistrates, either by random casting of lots or by senatus consultum. These appointments were not legally binding and could be overridden by the Roman assemblies.
Imperial provinces were governed by legates appointed directly by the Emperor and typically contained legions; senatorial provinces were away from the borders, rarely held legions, and had governors appointed by the Senate. The Emperor still held the power to override senatorial appointments and ranked above all other governors through imperium maius.
Egypt was treated as the personal possession of the Emperor rather than a normal province. Its governor, the praefectus Aegypti, held the highest-ranking equestrian post in the early Empire, and Egypt was an exception to the general rule against stationing legions in equestrian or senatorial provinces.
Beginning in AD 293, Diocletian reorganized the provinces into twelve dioceses, each overseen by a vicarius. Constantine completed the reforms by 318, removing military command from provincial governors entirely and organizing the empire into three Praetorian prefectures: Gaul, Italy, and the East.