I have two questions. I understand that if I specify the domain as .mydomain.com
(with the leading dot) in the cookie that all subdomains can share a cookie.
I'm not sure @cmbuckley answer is showing the full picture. What I read is:
Unless the cookie's attributes indicate otherwise, the cookie is returned only to the origin server (and not, for example, to any subdomains), and it expires at the end of the current session (as defined by the user agent). User agents ignore unrecognized cookie.
RFC 6265
Also
8.6. Weak Integrity
Cookies do not provide integrity guarantees for sibling domains (and
their subdomains). For example, consider foo.example.com and
bar.example.com. The foo.example.com server can set a cookie with a
Domain attribute of "example.com" (possibly overwriting an existing
"example.com" cookie set by bar.example.com), and the user agent will
include that cookie in HTTP requests to bar.example.com. In the
worst case, bar.example.com will be unable to distinguish this cookie
from a cookie it set itself. The foo.example.com server might be
able to leverage this ability to mount an attack against
bar.example.com.
To me that means you can protect cookies from being read by subdomain/domain but cannot prevent writing cookies to the other domains. So somebody may rewrite your site cookies by controlling another subdomain visited by the same browser. Which might not be a big concern.
Awesome cookies test site provided by @cmbuckley /for those that missed it in his answer like me; worth scrolling up and upvoting/: