While the US does not legally require a parental consent form, US Customs and Border Protection strongly recommends one. Here's the advice from their Web site, bolding mine:
Submit a Statement of Consent: Form DS-3053 or a notarized statement from both parents or guardians giving that person (example: grandparent) permission to apply for the child. Include a photocopy of the photo IDs for both parents or guardians.
If you are 18 or over – You will be able to travel abroad on your own without the consent of your parents. If you are under the age of 18 – It is up to the airline or travel operator whether you can travel abroad.
Legally or developmentally? Legally, it depends on your jurisdiction, however, by and large, children need the permission/approval of both parents to leave the general area of their home... To travel internationally, they need a passport and the consent of both parents, unless one parent has sole legal custody.
Unless there is a legal order that states a parent cannot leave the US or otherwise travel without consent, then it's not required. Taking a child on a trip or vacation is never kidnapping in the absence of a court order that states otherwise.
Use Form I-131 - Application for Travel Document. For permanent residents, the re-entry permit is valid for two years from the date of issue. For conditional permanent residents, the re-entry permit is valid for two years after the date of issue.
No. As a minor, you cannot enter the USA (or cross an international border of most countries) without being met, or travelling with a parent or legal guardian.
British-Brazilian dual nationals under the age of 18 who are travelling without both parents or legal guardians need authorisation from both parents or legal guardians to travel in Brazil or leave the country.