I understand that
Workplace translates into german as
Arbeitsplatz but in America we use variations such as "the office", "the shop", etc. And it seems that much of the context depends on the type of labor. Are their equivalents in German that are frequently used? Perhaps some to be more formal and others to be casual?
I understand that
Arbeitsplatz, as you already mentioned. However, that does not neccesarily mean workplace (in the sense i'd translate workplace, at least); i could say "Ich arbeite nun schon 6 Jahre am gleichen (Arbeits)platz" (which i would translate with workplace), but it could also be used in a newspaper headline "Die neue Fabrik schafft 130 Arbeitsplätze in der Gegend", which i'd translate to
Arbeitsstelle - a bit more formal, and with a slightly different meaning, more like "the company i'm working for". "Ich habe die Arbeitsstelle gewechselt" does always mean you changed your employer, while "Ich habe den Arbeitsplatz gewechselt" can mean you changed your employer, but might also mean you're just sitting at a different desk now.
Betrieb, Firma - if i come home late and my wife is mad at me, i'd say "Ich mußte länger im Betrieb bleiben" oder "ich mußte länger in der Firma bleiben", i wouldn't use either of the other words in that case.
Büro, Baustelle, Werkhalle, Laden, Backstube (bakery), ... - lots of words that refer to the building, or the room, i'm working in, so they don't translate to workplace, they rather imply workplace.
That depends on what your work actually is / where it takes place.
Every room with a desk in it might be called
If the office is just a (small) part of the workplace, one probably refers to the part which is most significant. For instance someone working on a construction area would rather refer to his workplace with
auf der Baustelle than
im Büro, even though most of the time he is actually working at the desk (even e. g. in a construction trailer/container or something like that).