Some Hungarian sentences can remove "filler words" because the context of the sentence and meaning are still intact without them
Yes,that's a reasonable translation, and I'd say report it if the question comes up for you again.
How does it come, that the same sentence can be singular and plural? Lawyer and lawyers are shown correct here. No distinction at all in hungarian?
In Hungarian, the sentence is definitely singular, but one could say Peter does not see a lawyer, Peter does not see any lawyers, or even Peter sees no lawyers, and they all mean the same thing.
Is it only for lawyers, or for nurses, teachers, animals a.s.o. as well?
There is nothing special about lawyers here. All these words in Hungarian have plural form that differs from the singular.
Because "the lawyer" implies a definite lawyer but the Hungarian sentence has no definite article in it.