If you want to go then there's no need to go on an organised trip, ferries for Tangier depart regularly from La Linea. But Tangier, while beautiful, was my least favourite part of Morocco due to the constant hassle on the streets. If you go be prepared for this!
I really enjoyed Gibraltar, though. Would that be another option for you?
drpends what you mean by "authentic". Is it Casablanca or Marrakech? Neither would be doable in 1/2 days from Andalucia (where are you staying?), the travel alone will take a day. I think you need more time or to visit Tangiers only.