Sure does. The manga version of Afro Samurai provides more depth and background to the world and characters. It's a great complement to the anime or standalone read.
Afro Samurai is a story about a young man named Afro on a quest for revenge. He is seeking to avenge his father's death. Afro has a unique sword and he battles many foes along the way. He has to face powerful adversaries in a world where the number one headband gives the wearer great power and is highly coveted.
The key elements are revenge, the number one headband, and Afro's sword skills. Afro is driven by revenge for his father's death, the headband is the symbol of power he's after, and his sword skills are what keep him alive in battles.
One great Samurai Jack fanfiction story is 'The Lost Years of Jack'. It delves into what Jack might have been through during his time away from his home. It's really interesting as it shows his struggles and how he managed to keep his spirit alive in the face of so many challenges.
It depends. Some consider them canon if they fit seamlessly into the established Samurai Jack universe and don't contradict the main storylines. Others might have different opinions.