As someone who has done some research in cosmology, I'll try to make some contributions:
1.) I agree with Nabeshin, that it might be better for you to have a major in physics. I work in the area of cosmological perturbations, mostly in the inflation era rightnow. I have the impression that most people who do research well in cosmology these days have solid background in both classical relativity and HEP. Thus having a broader spectrum in your knowledge will definitely benefit your graduate study. And yes I'm assuming that the physics majors will have advantages in this (though it could be a false impression since my major was physics).
2.) As Nabeshin pointed out, it's better to go to a department with more faculty doing actual research. Not only will this give you a tase of what research really feels like, you will also have more connections with people around the world who are studying the cosmos through seminars and workshops. Reasearch in reallity, especially for young guys, are not just what you do, but also what others are doing. Small departments aren't competetive here. And also if you changed your mind and decided not to continue with cosmology, which is verly likely, a department with faculty doing research in various sub-domains of physics, say plasma physics, could save you some pain.
3.) Now that you decided to go along with cosmology, there are a lot to learn. To understand inflation itself is some task that could be daunting as your readings of literature progress. It requires you to know QFT, statistical mechanics, statistics, some programming, and a lot more. And this is just inflation. Of course for a phd candidate, no one would be expeccting you to know all of these well. And I think leanring things thourgh research is a good thing.
4.) Say you finally made it through the 4-6 years of grad school, in your final year you would have to consider your chances of getting a postdoc after phd, as twofish-quant indicated. I'm not sure which country you are in, but here in the US, the chance of getting a postdoc for a fresh phd studying theoretical cosmology should be around 1%, and the positions are usually limitted to several places around the globe. If you did 2.) and 3.) well, namely you have a lot of interesting publications, you know lots of people in this area and your adviser can really help you get a postdoc position, your chances will be higher (but I'm not sure by how much).
5.) Doing some part time job while doing research is not impossible, but research itself is a demanding job. For theoretical work, you would have to sit down and do lots of calculations, integrals, and these could easily take hours before you can see any result. For experimental work, which I'm not familiar with, I guess pragramming for hours are also required.
All in all, what I can say is, don't just limit yourself to cosmology, since no real cosmologist does.
Hope these would help.