I don't know about PsyD programs, but at least for PhD programs, they want you to have the appropriate background (i.e. basic psych courses, stats) but doesn't require a psych major. Of course, this depends on the program itself. You can check out program websites and consult the graduate guide in clinical/counseling psychology book for more detail by school/program.
For the bolded part- again, it depends on the program, just like it is in medicine. Just have the pre-reqs done and some research experience won't give you a shot at acceptance at Hopkins or Harvard, but may give you a shot at lower tiered programs. Likewise, for clinical psychology, the more competitive programs will want you to have more than the required minimum. With that being said, there is also GPA, GRE, research experience quality, papers, posters, connections, research area, funding availability, and about 10 other factors that doctorate programs take into account. In my opinion, grad school admissions isn't as clear cut as it sometimes is in med school admissions.
PsyD programs, I would imagine, will place more emphasis on clinical experiences, but again, I never applied to PsyDs so I could be wrong. In general, I find that grad school (PhDs clin psych) don't care very much for "extracurriculars"- they mainly care about 1) research experience 2) research match. I think that grad programs place more emphasis on "fit" because you are (Again talking about phds here, not psyd) going to be working with one primary mentor. Fit doesn't just include how smart you are through your GPA/GRE but also, what are your career goals, what are your research interests, what kind of student you are, etc. etc. So I would say med schools place more emphasis on grades, whereas grad schools place more emphasis on the whole picture. That doesn't mean that grad school will accept you with a 3.2 vs med school accepting you with a 3.8. It just means-IMHO- that grad schools look past the gpa/gre (which, btw, will need to be good to begin with) into other aspects of your application.
As you can see, the answer is not as black and white.