Yes, we do think those rights should apply outside the US. Mainly because we've thought those were natural (or god-given, depending on preference) rights, not privileges provided by government, since our country's conception.
Actually not quite. The American Constitution is a contract between american citizens (aka The People) of what you promise not to do to each other. The US Government is not conceived of as an independent entity with its own identity but an emergent property of The People consenting to collect their rights together for the benefit of all The People, based on the pooling of their individual sovereignty. 'We The People' refers to American citizens.
Consequently, since people in other countries didn't sign on to The American Constitution, they haven't made any promises to you of which of your rights they wont violate and you have absolutely no expectation of your contract with your fellow Americans being honoured, also you are not bound by the Constitution to respect the rights of foreigners.
There is however an expectation that anything the American Government has promised to do towards foreign nations it will honour, because The People of 1 nation can freely enter into an agreement with The People of another nation, which is why American Treaties actually form part of the law of the land (and it says this in the Constitution). This, for instance, means the US government must honour the UN Universal Declaration of Human Rights inside the borders of any nation that is a signatory to it because the US is a signatory to it.
The bottom line is that the Constitution is a written contract between The People. The US government doesn't claim to be bound to always respect inalienable rights, but only whatever it expressly agreed to respect.
At the very most some foreign government can violate your so called inalienable rights and you could launch a civil lawsuit (or a revolution) against it for being wronged and a US court might agree with you. But nothing in the Bill of Rights claims that all of the rights contained therein are all inalienable rights.