How are international graduate applicants perceived by universities in the US? I've always wondered how professors and admission committees in the US think about international applicants for doctoral programs in the US. 

Skills:
Why are international applicants selected over (I believe sufficient) domestic applicants at all? Are there some skill differences?

Funding:
Isn't it more expensive and risky to provide >5 years funding to an international student than a domestic one? When do they think an international applicant trustworthy and appropriate for funding their studies?

I am wondering specifically about PhD programs (not Master etc) because some programs offer funding before even knowing the student in person which seems risky because if I were an advisor it would be difficult to evaluate a student's value for my lab before having actually worked with her.
To answer your specific questions, schools are generally trying to admit the best applicants (who will choose to come), not just a group of people of the right size over a certain threshold.  So if an international student appears to be a stronger candidate (evaluated the same way any other applicant would be), the international applicant is likely to be preferred.

Graduate students are generally funded based on packages that don't take into account where the student's from, so it's not generally more expensive.  (International students may be less likely to accept because the funding is less likely to enough, but that's no reason not to admit them.)

Finally, why would international applicants be any less trustworthy or appropriate than domestic ones?