Having a cookiless subdomain is largely un-needed unless you have either really large cookies or a very high traffic site. Having a cookiless domain for static content serving is quite common with high traffic sites as it does provide a measurable reduction in bandwidth usage and download speed, but this is only on high traffic sites. Low and medium traffic sites are generally better off doing site optimisation than worrying about cookies to start with.
As for how to implement generally cookies are set for the root domain and accessible to all sub domains by default unless you change where the cookie is attached to. In this case the cookie will still be a part of the request response cycle for a related sub domain. The most common method of having a cookiless domain is to have a completely stand alone domain for your static content which is different to your sites domain such as...
www.example.com and www.examplestatic.com
A good brief overview of the advantage of cookiless static content delivery can be found at StackExchanges own cdn which is https://cdn.sstatic.net/