Can you keep your foreign citizenship and marry an American?

Yes, My wife is Japanese and has her green card.

Yes you can.

I know many who have been married to US citizens for decades, live in the US, but for whatever reason have no desire to become US citizens.

Other than voting, running for office and a few travel restrictions Permanent US Residents  have the same rights and protections as US citizens. 

If bf had US citizenship as a child and gave it up when he was forced to chose just one citizenship, he is in a special category of people that are  eligible for permanent residence if they qualify (for example through marriage).  That is also why he is able to enter the US on a tourist visa.  You should have no problem. 

Japan doesn't allow duals which is why many Japanese immigrants to the US prefer to live as permanent residents so they can keep their Japanese citizenship. This is true for a lot of immigrants from any nation that doesn't allow duals. One can live a lifetime in the US as a green card holder. 

Yes.   To live permanently in the USA with their US citizen spouse the US citizen needs to sponsor him for either a fiancé' or a spouse visa.  Once that is done and he travels to the USA he becomes a US Permanent Resident. 

www.uscis.greencard

Do ask him what he did to renounce his US citizenship. Unless he went to a US Consulate outside the USA, paid the large fee, and formally renounced his US citizenship in the presence of a State Department official he is still a US citizen.  

Note that renouncing US citizenship usually results in the individual being permanently banned from the USA.  If he has visited the USA in the meantime then he is one of the exceptions, he didn't actually renounce his citizenship, or he was never a US citizen. 

Yes, your spouse could get a green card (permanent residency) and remain a Japanese citizen. There is no requirement for a us permanent resident to ever become a citizen. 

Yes you can marry anyone from anywhere... but marriage changes nothing else and if they renounced US citizenship they are not moving to the US to live, the option is they have to sponsor you to live in Japan

Sure.  I don't know of any country that bans citizens from marrying a person of another country.

Multiple citizenship is always a matter of all countries involved. Some countries allow it, some ignore it, India has a hybrid (OCI), and some countries treat it as a serious crime.

There are many people who hold 2, 3, 4, and even 5 citizenships, but Japan does not allow its citizens to hold another citizenship beyond the age of maturity. The United States doesn't care either way. Since your boyfriend gave up his US citizenship to retain his Japanese one, he will never be allowed to live in the United States again. If you ever were to get married and move to Japan, you eventually, after over a decade, would become eligible for naturalization in Japan. At that point the Japanese government would require you to renounce your US citizenship as part of the process.