I moved your question to the Graduate School forum because a PhD in psychology is more aligned with graduate school, not careers in medicine.
1.) No, not really - either OSU or OU will help you in your quest to get a PhD in psychology.
2.) No, that’s not quite correct. Number one, you don’t need a master’s degree to get a PhD - many people go straight from a BA to a PhD program. If you keep your grades high and get the kind of research experience you need, you don’t necessarily need a master’s. Number two, no, admission into the doctoral program of a department is not guaranteed if you earned your master’s in that department - although the likelihood does go up if you did a good job and impressed faculty.
3.) At the best PhD programs in both of your fields of interests, an offer of admission comes with full funding. Full funding means that your tuition, fees, and health insurance are 100% covered and you are offered a living stipend - usually in the $20K to $30K range, which will allow you to live modestly in most places. You never want to accept an offer of admission to a PhD program without full funding. They have nothing to do with financial need, but they do have to do with academic performance, your statement of purpose (in which you describe your research interests and why you want to attend a specific program), letters of recommendation and your GRE scores.
There is still a difference between in-state and OOS tuition, but as a graduate student you can usually gain residency after the first year of attendance. There aren’t any federal grants that graduate students are eligible for.
4.) It depends on what you mean by “best.” Some departments are known for being good overall, and some departments have strengths in specific subfields. Some examples of top departments of psychology (in no particular order) are UCLA, Michigan, Harvard, UC-Berkeley, Stanford, Princeton, Yale, Wisconsin, Brown, Minnesota, URochester, Columbia, CU-Boulder, Johns Hopkins, Carnegie Mellon, Pittsburgh, MIT, Iowa, Penn, Duke, UCSB, NYU, Penn State, Indiana, UVa, UF, UIUC, Oregon, Ohio State, Michigan State, Arizona State, Stony Brook, UNC, UCSD, Washington, Boston U, and Emory. But different ones are known for different things - Rochester and MIT are well-known for their cognitive psychology subfield; Minnesota has strengths in child psychology; Michigan and Stanford are strong all-around but well-known in social psychology; Arizona State has a strength in quantitative psychology. You’d have to do some individual research to determine which ones are best in the subfields in which you’re interested AND where there are professors doing research that interests you.
5.) Summa cum laude would be an advantage, but not because of the Latin honors - simply because that means you have a higher GPA, which is necessary to be competitive. A double major would only give you an advantage if it were relevant to your area of interest. A double major in sociology wouldn’t really give you any kind of advantage in social psychology - or any subfield, really - because sociology and psychology are distinct fields. (They have overlapping areas that they are concerned with, but they are quite different). But if you were interested in I/O, a double major in business might give you an advantage all other things being equal; or if you were interested in quantitative psychology, a double major in math could help you.
6.) More opportunities, probably. More debt, no - see #3.
If you want to be a professor, do note that psychology is a highly competitive field and academia is a field in which prestige matters. Those who are most competitive for careers as professors get their PhDs from top programs. Take a look at psychology departments of a variety of universities and see where they got their PhDs. You will see that top programs are more heavily represented, especially a big research universities and elite private liberal arts colleges. Even if you look at OU’s faculty, you see a lot of people from top programs - Yale, UT-Austin, Northwestern, Indiana, UCLA, UNC, Micigan, UConn, UC-Davis - as well as a lot of faculty from good solid mid-ranked programs - Buffalo, Florida State, George Mason, Texas A&M, UGA, OU, Colorado State, Binghamton. Some of these places have strengths in a field - like the professor from Colorado State went there for I/O psych, where CSU as a strength.
It’s generally stated in academia that you can teach at a university/college that is at your ‘level’ or down. So someone who got their PhD at Stanford or Michigan would be able to teach pretty much anywhere they wanted; but if you got your PhD at OU, you wouldn’t necessarily be able to teach somewhere like Michigan or Stanford - or even Rochester or Emory - you’d probably be looking at peer schools of OU. Likewise, you’d be unlikely to be as competitive for a slot as a professor at Pomona or Swarthmore, or maybe even Rhodes or St. Olaf, but perhaps at another small college like Austin College or Wesleyan College (in GA). You can improve your chances by publishing like mad, but the odds are still against you.
Outside of academia, it kind of depends on where you want to go and what you want to do. It does matter, but not nearly as much as in academia - especially because unlike in academia you can make up for lack of pedigree with experience and expertise.