I think if you are already in the US now, then it might be less important to stay in the US for the summer internship since you would have plenty of US connections from your undergrad. But it would still be good to have more connections in North America/US in general, if that is where you want to stay. I am from Canada and considered a PhD in the UK and Europe but among other reasons (difficulty of getting funding for non-UK or non-EU people), people advised me to stay in North America and build deeper connections if I want to end up with a job on this continent!
I think it would help to have something related to what your grad school plans are, but not absolutely necessary. But this might also depend on the subfield. In undergrad, I did 8 months work on a cosmology project, 8 months on medical physics, my thesis on astronomy/planetary science. I did a Masters on astronomy/planetary science, that is a bit similar to my undergrad thesis. My PhD program is also in astronomy/planetary science. So, I think having proven experience is helpful, but not all of your undergrad experience needs to be in the same subfield. In fact, profs that saw my PhD applications commented on the broad research background in a positive way.
I think you should do the research project that interests you more because that will also usually end up as better research. I think at the undergrad level, research experience itself is the most helpful thing, not necessarily the topic of your research. If you like what you do and want to continue at graduate level, then great. But if not, don't feel that you are only limited to applying to graduate programs based on your choice this summer!