Fall and spring internships are semester-long. I have a feeling a lot of that is rumor. So the myth that you must have connections to be chosen should not deter you from trying. Setting aside the well-connected applicants, there are certain things the internship office looks for in each application. It helps to have some substantive experience on your resume, but the amount expected varies according to your age.
Interns range from freshmen to seniors, though many are juniors.
The program directors look for demonstrated interest in politics - but based on the different kinds of experience my fellow interns had, there's no minimum requirement. In high school I volunteered for local politicians and I was politically active on campus, but I had not spent any time working or volunteering in Washington. Other interns had worked on international issues in the U.S. and abroad, and many studied political science or government in college and interned in that field while at school. But there were also interns who looked at politics through the lenses of other fields. In my case that was economics, but others were majoring in literature, sociology and psychology.