What do we know about the city of Capernaum? Jesus' ministry HQ

The exact location of the biblical city of Capernaum is not certain, but there are two serious sites on the north shores of the Sea of Galilee. Unlike Nazareth, Capernaum was referred to as an important city, which the mile-long archeological ruins support. The city flourished due to its proximity to a major East-West highway which crossed the Jordan river. Jesus spent an incredible amount of time in Capernaum, so much so that Mark refers to it as a place where Jesus was “at home” (Mark 2:1). It was in this city that numerous biblical accounts took place:

• Jesus sought refuge in Capernaum after the Jerusalem temple cleansing (John 2:12)
  
  
• Peter and Andrews' home was in Capernaum (Mark 1:29)
  
  
• Jesus's healing of the man with an unclean spirit took place in Capernaum's synagogue (Luke 4:31-38)
  
  
• Capernaum is the setting for the incidents and sermon following the feeding of the five thousand (John 6:16-59)
  
  
• Mark's account of paralytic's healing took place in Capernaum (Mark 2:1-12)
  
  
• Jesus also healed a Capernaum official's ill son (John 4:46-54)
  
  
• Jesus healed the servant of the centurion who build the synagogue of Capernaum (Matt 8:5-13)
  
  
• Jesus' discussions on true greatness also took place in Capernaum (Matt17:24-27)
  

Yet despite all the time Jesus spent teaching and performing miracles in Capernaum, the city still did not repent and believe, and is thus is listed among the cities of condemnation by Jesus (Matt 11:23, Luke 10:15).