"Też jestem w domu" is correct and has the same meaning as "Ja też jestem w domu" (the second stresses the "ja" more: "Ty jesteś w domu. Ja też jestem w domu." = "You are at home. I am at home too"). The sentence suggested by you, "Jestem też w domu", is grammaticaly correct, but the meaning changes (and therefore it sounds a little bit artificial): "Jestem w Polsce. Jestem też w domu." = "I am in Poland. I am at home too." (you are at the same time both in a country and in a building) - now "też" refers to "w domu" not to "ja".
Thanks for the clarification. We thought it was "right but sounds odd" as well (even my go-to native speaker wasn't 100% sure!)
Your example would say that you are at home, but also somewhere else at the same time, at least that's what logical stress imply. It's best to write it as "Ja też jestem w domu." (you can also drop the initial "ja").
So if I were to be very tricky, would it then be correct to say something like "Jestem terazw pracy. Jestem też w domu" if I work at home?
You missed the space between "teraz" and "w", but… yes, that's a great example.
Can someone please explain the difference between za, zbyt and też or is it just one of those things that sounds right or wrong to native speakers?
za/zbyt is for expressing excess. Like, there is too much of something, something is too long, too fast, too short, etc.
też is for when something happens or is together with something else. This use of "too" would be synonymous with "also", "as well", etc.
I would say that "zbyt" is slightly more formal than "za", but other than that, probably not much difference.