"This lady is not home."
Translation:Tej pani nie ma w domu.
The 'not at home' sentence has this weird, if translated literally, construction: There's no X at home. Seems understandable with stuff, like sugar, or computers, but I get it why it may be surprising with people. X is in Genitive then.
So this is literally "There's no this lady at home".
And "I'm not at home" would be in fact "There's no me at home". (Nie ma mnie w domu).
The second is "Nie ma jej w domu" = literally "There has not her at home". Sure, in English it makes zero sense. This construction is not exactly an intuitive one, I guess. Your "Ona nie ma w domu" means "She doesn't have at home..." - it misses the info about what it is that she doesn't have. So simply it's not what you wanted to write.
As for "Nie jestem w domu", that's not the most usual thing to say. True, it's correct, but it's like correcting someone that thinks I'm at home. "No, I am not at home, I'm actually at the post office. I cannot look for that envelope right now".
Meanwhile, "Nie ma mnie w domu" is more common and just states you aren't there. I'm 'missing from home'.