The name 'DefaultAuthenticationTypes' does not exist in the current context

For the first, it is just a constant string

If you do not want to install packages the quickest ( and not very orthodox ) solution is

ClaimsIdentity identity = new ClaimsIdentity("ApplicationCookie");

you can see definition here

DefaultAuthenticationTypes is part of Identity framework and found in Microsoft.AspNet.Identity namespace.

To use it, add a using to the top of the file

using Microsoft.AspNet.Identity;
//...other code
identity = new ClaimsIdentity(DefaultAuthenticationTypes.ApplicationCookie);

or call it directly

identity = new ClaimsIdentity(Microsoft.AspNet.Identity.DefaultAuthenticationTypes.ApplicationCookie);

The second issue was already dealt with in another one of your questions here

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