1. how much does it cost to register a domain name....
Depends on who you register it thru. I get mine at about $9.00 using www.domain-s.com
2. Is the registration fee same for a domain name of any type (.com, .org...etc) ..
See above, it all depends on who you register thru.
3. and does the amount vary with length of domain name too?
Again, depends on who you register with. Usually the longer you register it for, the cheaper you get it, etc.
4. In case I fail to re-register the domain name after the 1/2/x year contract-period and someone else tries to claim the same domain name ..does the fact that I owned that domain name give me an advantage to claim rights for the domain name back first?
Not unless you own some kind of trademark for that name most likely, other than that, you forget to renew and someone else grabs it, its theirs.