Here's a great overview of cookies in ASP.NET. But to more directly answer your question, cookies that you manage (e.g. that you create) are written like this:
Response.Cookies["myCookie"].Value = "Hello, World!";
and then subsequently read like this:
var myCookie = Request.Cookies["myCookie"];
// if myCookie == null it doesn't exist
and further, deleting a cookie is a variant of modifying them. It's a little weird, but to get the browser to delete it you need to set its expiration to something earlier than today:
Response.Cookies.Add(new HttpCookie("myCookie")
{
Expires = DateTime.Today.AddDays(-1)
});
But, if you're needing to read third party cookies, and you're trying to do that via ASP.NET, you would need to perform a cross-site scripting attack. There is a reason this is considered bad practice. This can be done via JavaScript, but I am not inclined to provide any code to do so because of its ethics.
EDIT
The OP updated their question while I was answering, and I do still stand on the fact that the intent of reading third party cookies via a web application is fishy at best.