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.
citizenship:
[...]
(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In any case, this information is only from a manual (and an outdated version of the manual at that), not from a law or even a regulation, so it is not very authoritative.
In the 2005 Digest of United States Practice in International Law, section "Citizenship Status of Child of Foreign Diplomat with U.S. Citizen Mother" (on page 1, which is page 33 of the PDF), it reported that a US embassy had made a request that year about citizenship of a child born in the US to a diplomat father and a US citizen mother, and the Department of State responded in a telegram that it is treated as if the child were born abroad, i.e. the child would have US citizenship if the mother met the conditions to pass on US citizenship to a child born abroad:
In response to a request from a U.S. embassy abroad for guidance in
determining the citizenship of a child born in the United States to a
foreign diplomat father with full privileges and immunities and to an
American citizen mother, the Department stated in a telegram that the
child’s “citizenship determination has to be made based on the same
rules as if he had been born abroad to one U.S. citizen parent.”
Therefore, if the mother “can document sufficient physical presence in
the U.S.,” her children “can be documented as U.S. citizens.”
Again, it does not explain the legal reasoning behind this.
The USCIS Policy Manual, in 7 USCIS-PM O.3(B), also mentions this in passing without specifying that the US citizen parent had to have met the requirements to transmit US citizenship to a child born abroad:
If one parent was an accredited diplomat, but the other was a U.S.
citizen or non-citizen U.S. national, then the child was “born . . .
subject to the jurisdiction of the United States,” and is a citizen.
Again, it does not explain the legal reasoning behind this.
There is another even more outdated reference on this. In the old INS interpretations on nationality law, Interpretation 301.1(b)(2), regarding the law on citizenship for children born abroad prior to 1941, contains this interesting sentence:
Under the above rules, a child acquired citizenship at birth in Puerto
Rico, 26/ Guam, American Samoa, Swain's Island, Philippine Islands,
27/ Alaska (before March 30, 1867), Hawaii (before August 12, 1898),
Virgin Islands (before February 25, 1927), the Canal zone (before
August 4, 1937), in any foreign country, and even in the continental
United States, 28/ provided, in this last instance, that the child had
an alien parent who was a foreign diplomatic officer duly accredited
to the United States.
The last part seems to describe the same principle (albeit before 1941) that children born in the US to foreign diplomats with diplomatic immunity can acquire US citizenship through the rules for children born abroad, even though the law before 1941 (like the current law) does not explicitly provide for that. Footnote 28 says "Application of Baron , CO 341-P (1961)." I can't find information on this; perhaps someone else can.
