Yes, the child is eligible. The child is automatically a Citizen of the United States under the Immigration and Nationality Act ("INA"), Section 320!
Yes they retain citizenship.
3 years of being permenent resident then you can apply
Good question; under normal circumstances, the alien woman will retain her prior citizenship(s), although she could probably apply for a Permanent Resident Visa if the marriage is recognized under Philippine law.
Yes.
No
Yes Citizenship Test is capitalized. The Citizenship Test is also known as the U.S. Naturalization Test and it is taken by those who are choosing to become permanent United States residents.
No, you will not be able to apply for citizenship while outside the U.S. Your husband will need to petition for a K-3 visa. Once in the U.S., you can apply for citizenship after being a permanent resident for at least 3 years.
Fulfilling the requirements for U.S. citizenship does NOT cancel your Canadian citizenship. Canadian citizenship is permanent, and (unless obtained by fraud) cannot be revoked by the Canadian Government for ANY reason unless you renounce it yourself. Also, the U.S. does not require Canadians to renounce their citizenship when becoming U.S. Citizens.
None of the member states of the EU will allow you to apply for their citizenship right after the marriage. You must first get the permanent residence of the concerned member state and meet all the requirements (e.g. residence period, language ability and even renouncing the US citizenship if they require you to). After that you can apply for their citizenship.
No, he is no longer married to a citizen so no longer has citizen status; unless he has applied for and completed the citizenship process to become a citizen in his own right.
No. Citizenship is not automatically granted to children born in Australia and people do not get automatic citizenship if they have a child in Australia regardless of whether they have a permanent visa or not.
Americans can live in Canada if they apply for permanent residence. They do not, however, need to "give up" their citizenship and may apply for a dual citizenship. They will have both a Canadian and their American citizenships.