How can I manually create a authentication cookie instead of the default method?

How can I manually create a authentication cookie instead of the default method? 

Here you go. ASP.NET takes care of this for you when you use the higher level methods built into FormsAuthentication, but at the low level this is required to create an authentication cookie. 

if (Membership.ValidateUser(username, password))
{  
  // sometimes used to persist user roles
  string userData = string.Join("|",GetCustomUserRoles());

  FormsAuthenticationTicket ticket = new FormsAuthenticationTicket(
    1,                                     // ticket version
    username,                              // authenticated username
    DateTime.Now,                          // issueDate
    DateTime.Now.AddMinutes(30),           // expiryDate
    isPersistent,                          // true to persist across browser sessions
    userData,                              // can be used to store additional user data
    FormsAuthentication.FormsCookiePath);  // the path for the cookie

  // Encrypt the ticket using the machine key
  string encryptedTicket = FormsAuthentication.Encrypt(ticket);

  // Add the cookie to the request to save it
  HttpCookie cookie = new HttpCookie(FormsAuthentication.FormsCookieName, encryptedTicket);
  cookie.HttpOnly = true; 
  Response.Cookies.Add(cookie);

  // Your redirect logic
  Response.Redirect(FormsAuthentication.GetRedirectUrl(username, isPersistent));
}

I'm not sure why you would want to do something custom here. If you want to change the implementation of where user data is stored and how users authenticate then it's best practice to create a custom MembershipProvider. Rolling your own solution and messing with the authentication cookie means a high probability of introducing security holes in your software.

I don't understand your part 2. You only need to call FormsAuthentication.GetRedirectUrl if you want to return users to the page they were trying to access when they got bounced to login. If not do whatever you want here, redirect to a url stored in the configuration if you want.

To read the FormsAuthentication cookie, normally you would hook the AuthenticateRequest event in a HttpModule or the Global.asax and set up the user IPrinciple context.

protected void Application_AuthenticateRequest(Object sender, EventArgs e)
{
    HttpCookie authCookie = Request.Cookies[FormsAuthentication.FormsCookieName];
    if(authCookie != null)
    {
        //Extract the forms authentication cookie
        FormsAuthenticationTicket authTicket = FormsAuthentication.Decrypt(authCookie.Value);

        // If caching roles in userData field then extract
        string[] roles = authTicket.UserData.Split(new char[]{'|'});

        // Create the IIdentity instance
        IIdentity id = new FormsIdentity( authTicket );

        // Create the IPrinciple instance
        IPrincipal principal = new GenericPrincipal(id, roles);

        // Set the context user 
        Context.User = principal;
    }
}