Claire S. answered 03/27/19
An English Tutor, for All Your Language and Literature Needs!
When you're applying for jobs and internships, cover letters are your chance to explain ahead of time why your experience and qualifications make you a good fit, what kind of connections you already have to the work being done, and provide insight into any gaps in your CV. Basically, it's your chance to highlight what you want, and explain anything you think might be confusing to the person reviewing your application.
In regards to your first question:
If you pursued certain academic courses or tests that relate to the research being done at the lab, that would be great to mention in your cover letter--it would also be a good thing to mention what drew you to that kind of academic research in the first place! But it's not necessary to go into detail on every academic result you've achieved, as they'll be able to read that in your CV. Ask yourself: what specifically shows I can do the work of their internship? What highlights the fact that my interests are similar to the lab's? Does any of my academic work pair well with what they're doing?
In regards to your second question:
It's a good idea to draw a connection between your goals and the professor's research. If you still have more academic coursework and research to do someday, you could talk about how your work with this professor and a specific aspect of their research will help prepare you for certain courses, certain degrees, or certain future research projects that you are interested in doing. By linking your goals with what research the professor is doing, it helps show that you're passionate about the work being done. However, you do want to avoid simply restating or summarizing the professor's work. You should put thought into analyzing how their research could have an important effect on your work and your education, to let them know that you're invested!