"He continues his job."
Translation:Il continue son travail.
If you read up on métier, I doubt that you would select that particular word to use for "job". "Travail" is your work-a-day thing; it's what you do to earn a living; it's where you go to do whatever it is you do to bring home the bacon. "Métier" is your profession. It's what you trained for. It's the field in which you have real experience and skill. But it may or may not be your "job".
I looked up "poste", "emploi", "métier" and "travail" and they all listed "job" as a possible translation. I'm sure the contexts are different, but since the sentences here in Duolingo do not provide ANY context - all of them s/b correct. I've submitted this for consideration.
I don't know which dictionaries you're talking about, but "métier" is not pejorative at all.
In your examples:
"Le métier de délateur." It's the job itself (informer) which is often badly perceived, a bit like tax collector can be ^^. The word "métier" doesn't add any negative connotation.
"un vilain métier" It's the adjective "vilain" which adds the negative connotation.
"métier" is a neutral word, it's the context which can make it pejorative or not.