The server does not keep track of cookies whatsoever. The webserver creates cookies, and sends them to your browser, at that point it forgets all about them.
When you visit a website, the webserver will ask your browser for any cookies that it has (for that particular server, not all the cookies on your machine), and your browser will send those cookies to the server. The server can then inspect them to find out any information that it had previously told your browser to save (such as the fact that you're logged in to a website).
Cookie expiration is handled by the web browser. When the webserver sends cookies to your browser, it gives expiration times for them, but there is nothing forcing browser developers to actually delete cookies at the scheduled expiration time. You could very easily write a HTTP client (a browser, basically) that forged cookie data.
You can think of a cookie like an ID card. A server gives you an ID card with specific data on it, including the expiration date for the ID card. Sometime later, when you come back to the server, it asks to see your ID card, if you have one. Your browser then gives that card to the webserver, so the server can see who you are.
There is nothing stopping your browser from forging an ID Card. They're plain text files, so if you know what data is supposed to be in them, you can edit them in a plain text editor. Also, if someone were to copy cookies off your computer, they could use them to gain access to websites you had logged into from any other computer they copied those cookies to.
But no, the server doesn't store cookie information, browsers do.