1. The most widely accepted form is without doubt " to be sitting at home". No definite article must be used.
Nonetheless it is frequent to hear some speakers say "I am sitting home" whereby they omit the preposition "at". The same thing also happens in cliches like "to go home" or "to be back home"
2. I didn't quite get the drift of the context. I mean, "he is tested" as to what? If you could be more specific about the context I would be able to aid you without much ado.
Student or Learner