How can I see the challenges of each club?
By going to the Map menu, there, under the icon below the icon of the map (shaped as an open book) you'll find all your "side quest", the clubs challenges, the explorers's missions, club quests, missions to acquire items for citizens and more.
Sure. Go to the Map screen and open up the Logbook. You will find a list of all the different factions and their respective challenges.