Where do the Cookie State and Session State information be stored?

Cookie Information will be stored in a txt file on client system under a 
folder named Cookies. Search for it in your system you will find it.

Coming to Session State
As we know for every process some default space will be allocated by OS.

In case of InProc Session Info will be stored inside the process where our 
application is running.
In case of StateServer Session Info will be stored using ASP.NET State Service.
In case of SQLServer Session info will be stored inside Database. Default DB 
which will be created after running InstallSQLState Script is ASPState.