No, the government has the responsibility to obtain the hostages by force or by POW exchange
No, paying such a ransom to terrorists means directly funding and suportting the killing of other people and encourages the kidnapping of other people.
The US gummint will pay the ransoms for individuals who are well-connected and influential within the corporate elite. It doesn't matter what the public thinks--the people have no say.
Yes. Their money, their choice.
Horse sense is the thing a horse has which keeps it from betting on people. ~W.C. Fields