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No, this is not dangerous. The traps are non-toxic glue. The only concern is ingestion of bits of the plastic tray causing foreign body problems.
Did he eat any of the tray?
No, it is unlikely that Marty got anything from the mouse itself. If you would like to be proactive, you can get him a dose of dewormer from your vet (like Profender). Aside from intestinal parasites, it is unlikely that he could have been infected by anything else such as rabies.