That is very strange. Looking at all of the network responses, there is no cookie ever set - all of the sudden it shows up.
Not in firefox though (confirmed)
Opera has a weird behavior, but only if redirected from /
I'm still looking into the other one. If I can find a way to re-create either of your issues, I will be able to figure it out - but as it stands, the cookie appears from nowhere. Plus, it's an HTTP cookie, so it can't be set by javascript - it MUST be set from an HTTP response, but when I review the HTTP responses from your site, there is no cookie set!