Your best source of information is the PMP Handbook. If you are going to go for your PMP take the time to review and understand what is required, doing so will help maximize your chances to succeed.
In the eligibility of the PMP certification it mentions "35 contact hours", I have searched on this for my local region and it seems to be that some training centers listed on the PMP site would provide with a certificate certifying these hours.
Is it really necessary to join a training center and get this certificate?
Can't I just study the books myself and appear for the exam?
Page 8 of the handbook specifies that the contact hours have to be "formal training", so intuitively NO you can't just self-study the PMBOK. It is not necessary to join a training center, there are any number of vendors, community colleges and universities out there offering pretty much any kind of training you may be interested in. Just make sure that they are "real" training organizations because there is a good chance you will be audited and if you don't have good documentation (i.e. some kind of certificate saying you took the course, the course lasted X hours and you passed) you will be out of luck.
4500/24 is 188 days - is that right calculation to go by?
Your calculation needs to factor in having a life outside of work in general and the need to sleep in particular. More reasonable is figuring that you work on projects ~5 hours per day (out of an ~8 hour workday) 5 days per week for 48 weeks per year (accounts for vacation, sickness, stat holidays), so you are looking at 45 months of experience.
If you are assume 100% efficiency over an 8 hour work day you are still looking at working full time at 8 hours a day you are still looking at ~28 months.
Also if someone is just leading teams chances are they are not yet managers (atleast by designation) and thus how would this 4years of managing projects be accomplished?
You can manage a project without being a manager per se. There is a significant difference between your role on a project and your title.