Learn Arabic (Modern Standard): رأيت رجلا أسود الشعر.

This is a special structure often called a "false Idafa" (an adjective + noun combination). Instead of saying "whose hair is black," Arabic creates a compound adjective by putting the adjective first (أسود), followed by the specific feature it describes (الشعر). It translates best as "black of hair."

رجلا is indefinite. Why is الشعر definite with الـ? Doesn't the adjective phrase have to match the noun?

The adjective phrase does match! In this structure, the definiteness of the whole phrase is determined by the first word, أسود. Because أسود doesn't have الـ, the entire phrase أسود الشعر is treated as indefinite, perfectly matching the indefinite noun رجلا. The word الشعر has الـ just to define the specific feature being described.

Since أسود is describing رجلا, why doesn't it end with the same double-fatha ("-an") sound?

Because أسود acts as the first part of an Idafa (a construct phrase) with الشعر. A strict rule in Arabic is that the first word of any Idafa can never take tanween (the "-n" sound). It still grammatically matches the "object" case of رجلا, but it only takes a single fatha sound at the end (أسودَ) rather than a double one.

Could I say this using a simpler structure instead of this compound adjective phrase?