The first Duke of Cornwall was Edward of Woodstock, the Black Prince. As a matter of fact, he became first the Duke of Cornwall (1337) and then the Prince of Wales (1343).
Only the Sovereign's eldest son can hold the dukedom, according to the charter of 1421. If the eldest son of the Sovereign dies, his eldest son does not inherit the dukedom. However, if the eldest son should die without children, then his next brother obtains the dukedom.
Underlying these rules is the principle that only a son of the Sovereign—never a daughter of a grandson—may be Duke of Cornwall. Thus, it is possible for an individual to be Prince of Wales without being Duke of Cornwall.
I do not know for sure who is entitled to the income from the Duchy when there is no Duke of Cornwall, but I think it is the British Sovereign, the dukedom having been reverted to the Crown.
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