You can sit for the CFP exam any time after completing the course work. You will still need three years of experience before actually becoming a CFP. If you pass the exam before you have the experience, the CFP Board requires that you still complete a certain number of continuing education credits each year.
"Fee only" planners only sell advice. If a planner is not "fee only," that just means that some or all of their compensation comes from some other source. To actually practice financial planning, you have to be registered, either as a Registered Investment Adviser (RIA) or as an Investment Adviser Representative (IAR), by a state government or the US Securities and Exchange Commission. This is like your license to practice financial planning. Many CFPs that work for brokerage firms are not actually allowed to practice financial planning because they and/or their firm do not have the the appropriate registration. Most of the major wirehouses do not permit their reps to become registered as investment advisers. Those brokerage employees are only allowed to offer advice if it is "incidental" to the sale of securities.
If you really want to practice financial planning, I'd suggest looking for a job with an RIA and not at a brokerage firm. Especially now, due a recent US Court of Appeals decision against the SEC, it will be even more difficult to gain meaningful experience working for a broker-dealer. I think most people go the brokerage or insurance route because they are unaware of the other options. The recruitment power of brokerage firms, banks and insurance companies is huge compared to the small, actual planning firms. If you complete the CFP course work, pass the exam but need the three years experience, you could still find a job with a planning firm as an associate planner or paraplanner while you gain experience. I think this is the best route to go if your ultimate goal is to be a practicing planner.
As for the degree you'd benefit most from, I agree with Bill, study something that interests or inspires you. Become a well rounded, educated person. I don't have any employees, but if I and when I do hire a planner it wouldn't matter if they majored in chemistry, English literature, or finance.
If you know that this is the career path you want to take, a good next step would be to start the CFP coursework. It amounts to 18 credits if you do it as part of your undergraduate degree requirements, so it could be done electives or a as minor if your school offers CFP approved courses. If not, perhaps a nearby school offers it. I believe most schools only require that your last 30 credits be from them. The CFP instructors can also offer you some career guidance and can help you begin building your career network. See if there are opportunities to intern with a planning practice near your home or school. You can also check into the Financial Planning Association's Residency Program, see http://www.fpanet.org/member/meetings/c ... 2997_1.pdf