Seneca was Nero's tutor, I think, not Gaius's.
He was I believe exiled, then pardoned by Claudius. I seem to remember being critical of the Claudian regime upon the ascension of Nero.
This is indeed true. Seneca, for Nero's amusement and to satisfy his own personal grudge, wrote a pamphlet on Claudius' fate after death, entitled 'The Gourdifaction of the Defied Claudius'. The title itself was a pun, meaning that although Claudius was declared a god, all he was really fit for was to become a gourd, a flavourless, hollow vegetable similar to a pumpkin.