Hi,
I just finished my Part III exams a week ago. I graduated from Southampton with a BSc in Physics. To answer your questions:
I think the minimum requirements on their website to even apply is a 1st class, so I assume they look for people with quite high firsts, around 80% though I know of people who have been accepted with lower grades.
In my undergrad I did tailor my degree towards more math modules that were relevant towards theoretical physics - and let me tell you that knowing more maths really pays off on this course. Also, make sure you do a BSc/MPhys project in theoretical physics or whatever is close to what you plan to undertake in Part III, I think it looks good on the personal statement and gives you a referee who can write about your potential in the area.
I'm not sure if individual marks are looked at our not, but best to try and do well across the board and I think it probably does help if you score highly in modules related to what you want to concentrate in during Part III.
Again, I don't know how interested Cambridge is on extracurricular things, but and internship and summer research can't hurt your chances.
The application process is simple - fill out the form and provides references and a personal statement, there won't be an interview.
Just a couple of extra notes/tips: The course will be much harder than anything offered at Southampton, there were no 'easy' modules - all of them required a lot of work to get through and understand. This is reflected in the high level of the students attending the program. I highly recommend if you are successful in the application to make sure you brush up on some basic Group theory, Differential geometry and Complex analysis, I wish I did prior to coming.
If you have any other questions, feel free to ask and good luck!