Is it safe to use such code?
Response.Cookies[cookieName].Path = Request.ApplicationPath + "/";
I want to know about all corner cases, please...
In short, no, it's not safe. Using cookie paths is fraught with problems as they are case sensitive in IE and Chrome, but not FF. This means any mismatch in path case will stuff things up.
When generating a cookie, if the path you set differs in case from what the user typed, browsers won't store it.
When the user returns, if the path they enter differs in case from the first trip, the browser won't supply the cookie with the request.
What problem are you trying to solve?