7 FAM 1100, archived here from an old version of the Foreign Affairs Manual from 1995, goes into detail about issues regarding birthright citizenship for children of foreign diplomats. See 7 FAM 1116.2-(2,3,4), on pages 7-10 of the PDF. Unfortunately, later versions of the Foreign Affairs Manual no longer contain this information.
The part that is relevant for your questions is in 7 FAM 1116.2-2(d)(4):
d. As a rule, children born in the United States to the
following employees of foreign governments acquire U.S.
(4) Diplomatic agents who have the children in question with
U.S. citizens capable of transmitting U.S. citizenship to
children born abroad. Such children acquire citizenship under
pertinent law as if born abroad and would be subject to any
citizenship retention requirements in effect at the time of birth;
This basically says that a child born in the US to one parent with full diplomatic immunity and one parent with US citizenship would basically be treated like a child born abroad, and would acquire US citizenship at birth if the US citizen parent meets the conditions for transmitting US citizenship to a child born abroad. So the answer to your question #1 is sometimes they have citizenship, and sometimes not. The answer to your questions #2 and #3 are implied to be no citizenship (since permanent residents and nonimmigrants cannot transmit US citizenship to a child born abroad).
I am not sure what the legal basis for this rule is. Why would a child born to a parent who can transmit citizenship be subject to US jurisdiction, while a child born to a parent who can't transmit citizenship be not subject to US jurisdiction? Or perhaps they are both not subject to US jurisdiction, and the citizenship derives from the section of law on citizenship for children born abroad to US citizen? But that section specifies "a person born outside the geographical limits of the United States and its outlying possessions", so it doesn't seem to apply to children born in the US. Or perhaps they interpreted that Congress didn't intended for children born in the US to have any less preferential treatment for acquisition of citizenship than children born abroad in the same situation, so if a child born to a US citizen and foreign diplomat would acquire US citizenship when born abroad, the child should acquire US citizenship when born in the US too.
In any case, this information is only from a manual (and an outdated version of the manual at that), not from a law or even a regulation, so it is not very authoritative.