Please see and upvote Eric's helpful answer. I have edited this answer to correct it in light of that information.
Question 1: is it REQUIRED that my son attain an American passport in order to go to the US?
If your son is a US citizen, yes and no. There is a law that makes it "unlawful" for a US citizen to leave or enter the US without a valid US passport; however, there is no penalty for violating the law, and a US citizen cannot be refused entry to the US. There are some anecdotal reports at Travel that people have done this sort of thing before with only a little hassle. If I can find some specific examples I'll edit this answer to add them.
I say if your son is a US citizen, because that will only be the case if you meet the requirements of both 8 USC 1401(g) and 8 USC 1409(a). Section 1401 requires you to have been
physically present in the United States or its outlying possessions for a period or periods totaling not less than five years, at least two of which were after attaining the age of fourteen years
(There is some additional text about including periods spent outside the US in military or government service, or in service to an international organization in which the US participates, such as the UN, as if they were spent inside the US.)
So if you don't meet the physical presence requirement, your son is not a US citizen, and you don't need to worry about getting a US passport. The rest of this answer assumes that you do meet the physical presence requirement
Section 1409 requires
(1) a blood relationship between the person and the father [to be] established by clear and convincing evidence,
(2) the father [to have] had the nationality of the United States at the time of the person’s birth,
(3) the father (unless deceased) [to have] agreed in writing to provide financial support for the person until the person reaches the age of 18 years, and
(4) while the person is under the age of 18 years—
(A) the person [to be] legitimated under the law of the person’s residence or domicile,
(B) the father [to acknowledge] paternity of the person in writing under oath, or
(C) the paternity of the person [to be] established by adjudication of a competent court.
So, unless you have made the written agreement required under number 3, your son is not a US citizen, and, as Eric's answer suggests, you can happily travel to the US with the German passport and ESTA. More information about the requirements for the written agreement may be found in the USCIS Policy Manual at Chapter 3 - United States Citizens at Birth (INA 301 and 309).
Question 2 should be asked separately, so I won't answer it here (also because I do not know the answer).
Question 3: if someone is a dual citizen, must they always possess a passport from both countries while traveling internationally?
No. There will be specific circumstances where some dual citizens will need both passports, but there is no general requirement. For example, I believe Poland has a similar requirement to the US, so a Polish/US dual citizen needs both passports (at least nominally) to travel from Poland to the US or vice versa. Most countries refuse to give visas to their own citizens in foreign passports, so a citizen of two countries that have a mutual visa requirement will need both passports.
For travel that does not involve both countries of nationality, however, only one passport is generally necessary. This can also be true if the country does not require its citizens to use its passport to cross the border. For example, Canada explicitly allows US-Canadian dual citizens to enter Canada with US passports, although it does discourage this. Similarly, a US/German dual citizen can travel between Canada and the US without a German passport, or between Japan and Australia without a US passport.
They both have German passports, and are prepared to attain an ESTA.
Apply for the child's ESTA authorization now. When doing so, you will have to declare that the child is a US citizen. We have seen on Travel that the US has granted ESTA authorization to dual citizens after they disclose their US citizenship, but that could change at any time. If ESTA authorization is denied, you will be unable to fly to the US without getting the child a US passport.
Also, do not fly to the US through Ireland or through any other preclearance airport without a US passport for the child. Someone left a comment on Travel suggesting that US preclearance officers will refuse to preclear US dual citizens without a US passport, in contrast to the practice at actual ports of entry. While this seems far less likely to happen to a baby, I wouldn't risk it if I were you.
