I was just about to say the same things. My D goes to Rice and it fits what you are looking for very well in terms of environment. The campus is beautiful but in an urban environment, no greek system, friendly students, honor code, etc. I’d also encourage a visit if you can manage it. As Houston1021 said, the overnight visits in the fall are popular.
I would caution you to think carefully about your major if you plan on pre-med. I sat in on a talk by the head of BioE at Rice and she commented that they do not encourage pre-med students to choose that major. You obviously want something you’re interested in but at many schools it will be hard to maintain the GPA you need for med school with the CS and Engineering curriculums. But that’s a whole other discussion!