What would have happened if Luke joined Darth Vader in TESB?
I think Vader would have taken Luke to Coruscant if Luke didn't jump down on Cloud City, to kill the Emperor, and then Luke either becomes Emperor Vader's second-in command Sith apprentice, or they rule the galaxy as equals (Emperor Vader and Emperor Luke). Either way, Luke becomes a Darth (something.), sith lord.
I would think the emperor would sense something eventually more so given he seems the type to exercise caution with Vader in regards to his force sensitive son. If they don't have to fight their way to the emperor (crimson guard blah blah) then I can see it playing out with them slaying him. Neither alone I think could do it (vader weak against lightning and Luke just lacking exp) but together they could do it. More so with Vader giving his son training on top of Yoda's.
Darth Vader being a family man would share power with his son and become the new emperor. Luke his right hand filling in vaders old position but with more authority. Leia would have to be killed..(if so then by Luke) She's vaders last weak spot and I doubt she would turn to the dark side given her beef with the death star blowing up her planet. If Luke did go with vader I would imagine any attempt to liberate Han Solo would end horribly. Perhaps Luke+Vader come to the rescue to save her if not show up after Boba (after collecting his bounty on Han solo) moves onto his next job which is liberating the princess leaving Han a coffee table and the Wookie to fight the rancor.
She wouldn't join the empire and be locked away or used as a decoy to lure out the rebel scum.
Agreed. The Emperor wanted Luke to take Vader's place at his side. Vader even says, "Luke, you can destroy the Emperor. He has foreseen this. It is your destiny. Join me, and together we can rule the galaxy as father and son" I'm sure Palpatine gave the same line to Dooku, but in the end, he wants Luke to kill Vader and become his new apprentice.