Nigel tells Bug he hasn't spoken to his father in a half-dozen years. While true that Nigel had a bad relationship with his father, in "Murder in the Rue Morgue" (season 4 episode 11) he told Jordan that he and his father made up and were back on speaking terms.
Both Woody and Nigel call the T-33 a fighter jet. The T-33 was a trainer, albeit based on the F-80 Shooting Star fighter.
Nigel and Bug refer to the T-33 as a fighter and that it had a navigator. The T in T-33 stands for trainer and did not have a navigator. It would have been more realistic to refer to the aircraft as an F-94, an interceptor aircraft based on the T-33 which had a Radar Intercept Officer in the back, similar to, but still not a navigator.