There's the Charles Darwin Research Station which, if memory serves, only displays Galapagos tortoise and Galapagos land iguana. It's conservation work is more impressive than its exhibits, although they're not bad by any stretch. As for the pet trade, there are incidents but I don't think it's one of the major threats to Galapagos wildlife.
Sure. These are the two most recent cases I know of: German convicted of smuggling Galapagos iguanas Man arrested for smuggling Galapagos iguanas | Galapagos Conservancy If you can ignore their other activities, there's some more info on the Sea Shepherd website: Sea Shepherd Galapagos Wildlife-Sniffing K9 Officers Bosco, Cristina, Kevin and Nico Retire - Sea Shepherd Conservation Society