You are correct about the idea of dual citizenship being recognized by other countries.
The US makes it illegal for an American citizen to seek citizenship in a foreign country, vote in a foreign election, or serve in a foreign military.
Not exactly.
If a US citizen takes on foreign citizenship without the intent of relinquishing US citizenship, US citizenship is not lost. The presumption is that the American citizen does not intend to relinquish US citizenship, and if the question arises, the individual is asked by a State Department official. If the individual states that he did not intend to relinquish US citizenship when he takes foreign citizenship, the State Department official certifies in writing that the individual did not so intend, and that's that.
It's not a crime to take foreign citizenship, nor to serve in a foreign military, nor to vote in a foreign election (unless one holds a position of trust with the US government, in which case it may be, but that is a separate case). What doing those things does, is creates the possibility of relinquishing US citizenship. However, the legal presumption is that one did not so intend.