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.
You will never amount to much. -- Munich Schoolmaster, to Albert Einstein, age 10