"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".
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.