Which part of Excel Services is responsible for maintaining Session States?

The Excel calculation Services.
Excel Calculation Services maintains a session for the duration of interactions with the same workbook by a user or caller. A session is closed when the caller explicitly closes it or when the session times out on the server. Excel Services caches the opened Excel workbooks, calculation states, and external data query results, for improved performance when multiple users access the same set of workbooks

What is Excel Calculation Service? Explain the application server layer?

It is hosted on the back end.
1.    The main responsibility of ECS are :
2.    Loading workbooks
3.    Making required calculations
4.    Calling previously coded user defined functions[UDFs]
5.    Maintaining Session States for user interactivity.
Note: All of this is done using Sharepoint Shared Service Provider.

Explain the architecture of the Excel Services?

Excel services is built on ASP.NET and Microsoft Windows Sharepoint technologies.
The main components of Excel Services include :
1.    Excel Web Services
2.    Excel Web Access[EWA]
3.    Excel Calculation Server [ECS]
They are basically grouped into two layers :
1.    The Web Front End [WFE] where Excel Web Access [EWA] and Excel Web Services [EWS] reside.
2.    The application server where Excel Calculation Server [ECS] resides and where all the calculations are made.

Does Excel Services supports all set of features which Excel application on the desktop supports?

No.Only the subset of features from Excel application are supported.

Does any Office clients or viewers or ActiveX control installation is required for Excel Services?

No

In which edition does the Excel Services is included in Sharepoint?

The Eneterprise Edition of Microsoft Office Sharepoint Server.

Excel Services

It is a services designed to give the users the ability to easily and securely share Excel Spreadsheets and workbooks.