Well, you got two questions.
Is it possible to implement authentication and authorization trough session variables without using forms authentication and forms authentication cookie stored in browser?
yes it's possible but we're not supposed to reinvent the wheel especially it is related to security. It's strongly recommended to use form authentication when possible unless you have strongly valid reasons.
How is session id being sent in consecutive requests? Is it stored also in a cookie? What are the drawbacks of using session instead of forms authentication for authentication?
to see the cookie.
step1: Create a new ASP.NET MVC project using internet template.
step2: Start it and create a new user and login.
step3: Open developer tools check the cookie section you can see two cookies
__RequestVerificationToken
.ASPXAUTH
.ASPXAUTH is the cookie that FormAuthentication consume to do the authentication. For all following requests to the server, the server will check this cookie to authenticate user.
You can specify "Remember me" when you login which will changes the life span of this cookie, if you don't tick it the life span is tied up to current session, if you tick it depends on the settings on the server side.