Robin would be constantly thinking of escape plans. He'd use his knowledge of martial arts and acrobatics to his advantage. If he's in a room, he'd look for vents or other possible escape routes. And he'd be biding his time, waiting for the perfect moment to make his move, all while keeping his cool and not showing any fear to Slade.
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.
Well, in a fanfiction where YJ Robin is kidnapped by Slade, one possibility is that the other members of the Young Justice team would immediately start a rescue mission. They would use their unique skills and powers to track down Slade's hideout. For example, Aqualad might use his connection to the water and sea life to gather intel, while Superboy could use his strength to break through any barriers in their way. Robin himself would likely be using his wits to try and outsmart Slade from within the captivity.
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 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.
In this kind of fanfiction, Robin might be in a lot of danger. Slade could be using Robin to get to Batman. He might try to brainwash Robin and turn him against his friends. For example, he could lock Robin in a secret lair and show him false information about Batman to make Robin question his loyalty.
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 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.
Batman, of course. He's Robin's mentor and protector. He has the skills, the resources, and the determination to track down Slade and rescue Robin.
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.
Yes, there are likely some out there. You can start by looking on fanfiction websites like Archive of Our Own or FanFiction.net. Just search for 'Slade hurts Robin' and you should be able to find relevant stories.
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.