I wanted to know if its possible to do the following:
- Buy Domain from GoDaddy.
- Buy Hosting from HostGator.
- Buy DNS Management from DNS Made Easy.
Yes you can. If you look at this article on the Godaddy website: http://support.godaddy.com/help/article/664/setting-nameservers-for-your-domain-names and look at the section "Setting Nameservers for a Domain Name Registered Elsewhere and Using Off-site DNS" it will tell you how to set up the DNS. Then you need to configure the DNS to point to your hosting.
Yes! It is possible to register a domain name from Godaddy, purchase a hosting package from hostgator and use the DNS management from a different company.
It is advisable to use the DNS management service provided by your hosting company itself; Hostgator in this case.