Unless you're a PhD student with a unique skillset, without authorization to work in the US you will not be able to get anywhere. It takes a lot of time, effort, and money for an employer to sponsor someone and it is highly unlikely any company will do that for a mere intern.
Generally internships with medium to large companies pay well, almost as much as an entry level full time employer. A small company or startup may not be able to afford that.
I disagree. It depends on the company. The fortune 500 company I work for accepts foreign students and jumps through the hoops to get them work visas even if they are only undergraduates.
"The fundamental principle of science, the definition almost, is this: the sole test of the validity of any idea is experiment." -- Richard P. Feynman