Incorrectly regarded as goofs
Newt actually states that the cats are "spirit familiars", which may be something very different from the traditional concept of familiars that Rowling stated do not exist in the Wizarding World.
It's already been established, in that same film, that there are exceptions to the prohibition on apparation on Hogwarts grounds. I would expect that if the Hogwarts Headmaster has the ability to circumvent the ban, it's reasonable to guess that the Minister of Magic has a similar exception.
Dumbledore having taught Transfigurations does not preclude him from also having taught DADA at one or more points during his tenure. There is precedent for a particular professor having been associated with more than one subject.
Actually this is referring to the emblem of the Lestrange family. The family crest is a raven.
In dialogue, it is implied that the Thunderbird mission was a cover for Newt's actual mission for Dumbledore. It's certainly possible that both missions were considered to be of equal importance to Newt.