I am assuming your son is under 18 years of age.
Yes, United States citizens can enter the asylum procedure. Their applications will generally be rejected as the United States is considered a safe country. It is correct that it is not possible for him to receive a residence permit while you are as a family still in the asylum process. You have to await the results of your case. If it is positive, then he will be entitled to a residence permit as the child of someone with a refugee status. If it is negative and you are entitled to a Duldung, he will also be issued with a Duldung.
Regarding your case, it is not clear to me what kind of lawsuits you are referring to. You mention a court, the administrative court and a lawsuit ‘to know what kind of steps they have taken’. I would need to have a clearer picture of your situation in order to provide an answer on it.