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Yes dual citizenship is allowed for US citizens.
Yes. If one takes up citizenship in another country, then the US will consider that person to no longer be a citizen. There are circumstances where you can hold dual citizenship if you were born of a US parent in another country and vice versa. Many Canadians and US citizens are dual.
The US recognizes dual citizenship.
Ted Cruz renounced his dual Canadian citizenship.
No. Justice Elena Kagan is Jewish, but is a US citizen born to US citizens. She does not hold dual citizenship in the US and Israel.
I think that if your father was (and/or still is) a US citizen, then you are also a US citizen no matter where you were born. It should be rather easy to get an american passport in the US embassy or consulate in the phillipines for you. As for your mother, I believe that she has to apply for a US visa or "green card" based on her marriage to your father who was/is a US citizen, or thru your US citizenship if you are over 21 years old.
No. The spouse would have to naturalize in France to become a french citizen. Children of this couple, who are born in the US, are dual French-US citizens at birth. The process for "recognizing" french citizenship is complicated, however.
US has not stopped dual citizenship. It was banned in the 19th century, however, due to changes in law, it was again revised in the 20th century. It is only allowed at considering certain circumstances. America considers many factors before issuing it to the individual irrespective of the country born. Individual's citizenship is based upon the parent's origin country. Suppose a child is born outside US, parents are US citizens, then the child will be given dual citizenship one from the country born and US citizenship based upon parents. Naturalization is also common nowadays for getting dual citizenship.
You can apply for dual citizenship to the USA and almost any other country. Many more people than you might think hold dual citizenship. But the road to dual citizenship, with only a few exceptions, is long, twisting and full of "gotchas".
she have to marry again in israeli
By law, the US doesn't recognize dual citizenship (i.e. constitution prevents dual nationality) but somehow it always does recognize it.
No. The US president MUST have US citizenship only.