Yes, summer programs are ok.
Yes, undergrad research counts.
Yes, an NIH research project may help your application.
Typically, medical students engage in research during the summer and winter breaks between first and second years. A few have significant work done before medical school and should list that on their applications.
The best research is long-term, medically relevant and results in a publication in a peer-reviewed journal (first author, preferably). Barring that (and having that is pretty unusual for most medical students), something you find interesting, preferably related to your future residency field and can discuss in detail during interviews is best. "Name brand" research projects (ie, an NIH grant or summer program) do not inherently improve your application if your duties were limited to Chief Cook and Bottle Washer. Learning the principles of research and making a significant contribution to the work, is, IMHO, more important than working at some high-powered place. That said, the more people you meet while doing research (and impress), the better off you will be when application time rolls around.
Hope this helps.