While I'm not able to discuss the technicalities of 14th century armor (edit: see /u/WARitter's excellent response for all that and more), I can address the nickname "Black Prince". It was not contemporary to Edward's lifetime and it's origins are not known for certain. He was known sometimes as Edward of Woodstock (his birthplace) though more often as the Prince of Wales, a title which had belonged to the English king's heir apparent since 1301. Chronicles contemporary to Edward's later years refer to him as “Edward the Fourth”, and this was also commonly used to refer to him after his death, until 1461 when the crowning of an actual Edward IV made such usage obsolete.

The first written use of "Black Prince" comes from the mid 16th century "Collectanea”, an anthology of notes, transcripts, and summaries of earlier British writers compiled by John Leland. In his summary of a chronicle by John Warkworth, Leland writes of how “the blake prince wan a Feld of the frenchmen by Chaveney”. In his summary of a latin eulogy to Edward, Leland names him as “Edwardi Principis cog: Nigri”. However, in both cases the usage is Leland’s. It is not present in the original manuscripts he summarizes. What this suggests is not that Leland invented the term, but that by the time he was writing the Collectanea the nickname “Black Prince” had sufficiently attached itself to Edward’s memory that Leland could feel comfortable referring to him by this name.

Of course, none of this addresses where exactly the name came from. Black armor has indeed been the most guessed at reason, but I'm unaware of such armor being described during Edward’s lifetime. Richard Barber, in his biography of Edward, suggests the name’s “true origin may be in pageantry, in that a tradition had grown up of representing the prince in black armor.”

Source - Richard Barber "Edward: Prince of Wales and Aquitaine"

So the unanswered question remains... how would he have made black armor and was black armor common or is this just fiction making things more interesting?