Robin might initially be defiant. He's a trained fighter and detective, so he'd likely try to gather as much information as he can about his surroundings. He could be looking for any weaknesses in Lex Luthor's security setup or trying to figure out what Luthor's endgame is.
Robin could also be trying to communicate with his allies in some way. If he has any hidden gadgets on him, he might be attempting to send out a distress signal. He'd be constantly on edge, aware that Lex Luthor is a dangerous adversary. He may be mentally preparing himself for any sort of interrogation or torture that Luthor might subject him to, and thinking of ways to resist and not give up any important information.
Robin would probably be both scared and angry. He's a brave kid, but facing the Joker alone is no small feat. He might initially try to fight back if he has the chance.
A more unorthodox plan could be for Oracle to spread false information about a powerful new weapon that only Robin knows how to operate. This would make Luthor believe that Robin is more valuable alive than dead. Then, while Luthor is distracted trying to figure out how to use Robin to get this weapon, the heroes could stage a rescue. They could also try to turn one of Luthor's minions against him. Offer the minion a deal in exchange for information on Robin's whereabouts.
Robin would probably stay calm and observant. He's been trained well, so he'd immediately start looking for any weaknesses in his captivity situation. He might try to pick the locks on his restraints if he's bound, or find a way to communicate with his team secretly if there's a chance.
Robin could use his intelligence to analyze Deathstroke's lair and find a weakness in its security system. He might notice a pattern in the guards' patrols or a flaw in the locking mechanisms. Then, he could use his acrobatic skills to create a distraction and slip away unnoticed.
The other Teen Titans could also play a big role in the rescue. Starfire could use her powers to fly in and distract Slade while Cyborg and Beast Boy create a diversion. Raven could use her magic to break any spells or traps that Slade has set for Robin. They would all work together as a team to get Robin back safely.
Robin would probably be very hurt. He might initially go into a state of shock and disbelief. After that, he could try to find evidence to prove his innocence. For instance, he might search for surveillance footage or eyewitnesses.
Superboy could use his super - strength to break free from whatever physical restraints Lex Luthor has put on him. If he's chained up, he could simply snap the chains. Then he could use his super - speed to run out of the facility before Lex Luthor's guards could react.
Beast Boy would be quite distraught. He might initially feel a bit helpless, but then he'd start thinking of creative ways to find Robin. He could transform into various animals to search for clues or to sneak into Red X's lair. His sense of humor might also help to keep the team's spirits up during the search, even though they're all worried about Robin.
It could be Nightwing. He was once Robin himself and has a special connection to the current Robin. Nightwing might hear about the kidnapping and step in to save Robin, using his own unique fighting style and experience.
Well, Lex Luthor's death varies depending on the comic storyline. Sometimes it's due to his own schemes backfiring, other times it's at the hands of a powerful foe.