It could be the Joker. He's always up to no good and kidnapping Robin would be a great way to mess with Batman. The Joker might want to use Robin to force Batman into a series of cruel and twisted games.
Well, one possible plot development could be that the kidnappers demand a ransom from Batman. Batman then has to race against time to find Robin without alerting the media or causing panic in Gotham. Another development might be that Robin manages to leave some hidden clues for Batman to find while being held captive. Maybe he uses his martial arts skills to briefly overpower one of the kidnappers and mark a symbol on the wall that only Batman would understand.
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.
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.
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 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.
In the world of fanfiction, there are often hidden gems. For 'Robin kidnapped' fanfictions, you might find good writers by looking at the reviews and ratings of stories with this plot. Writers who get a lot of positive feedback for their take on this theme are likely to be good. They may be skilled at character development, making the story of Robin's kidnapping and possible rescue really engaging.
One idea could be that Poison Ivy kidnaps Robin to use him as a bargaining chip against Batman. Maybe she wants Batman to stop interfering in her environmental plans. So she holds Robin in her lair, which is filled with all kinds of dangerous plants.
In many fanfictions where Gil gets kidnapped, it could be for various reasons. Maybe it's by a rival group who wants to use him as a bargaining chip against his friends. For example, in some stories, Uma's crew might be targeted because they are seen as a threat in the Isle of the Lost. Gil, being a part of that crew, gets kidnapped. Writers often use this plot device to create drama and show the loyalty and courage of the other characters as they try to rescue him.