Ok registered now and not sure why got a -1 vote on previous answer but will expand.
Web Developer Express 2008 Does not create or work with Sql Server CE simply because SQL Server CE is not designed to work on the Web Server.
When you load c# express or vb express. sql server express 2008 is offered to be installed. SQL CE isn't installed. if you go and add a data source the two I know will be there, Microsoft SQL Server Compact 3.5 (.NET Framework Data Provider for Microsoft SQL Server Compact 3.5)and Microsoft SQL Server Database File (SqlClient), the access client may be there also.
It should be noted with a default Express setup that if you choose to use the data provider for sql server compact it will fail even though it is an option. you would have to download sql server ce separately to create sdf databases.
Also Express versions are targeted to a different audience and doesn't have the same ui setup as the main vs sku's have. For example Web express in database setup doesn't even offer the choice to use sql server ce. while in VS it is a choice from the data menu mainly because VS supports more than just web developer.
Now you said you loaded VS 2008 (didn't say what version but doesn't matter since this affects standard and above)
The default install for vs 2008 will install Sql server ce (and as previous answer stated 3.1 was with rtm, 3.5 was with sp1). you can turn it off in advance options.
One concern I have is that you mention that you can't create a database in vs on web projects. which template are you using. You should be able to create server based database Items in add item and both local and server database objects in the data tab.
so in Summery.
on client based express skus and client based projects you should be able to create a local or server based database from add item and from the data tab.
Since sql CE is not officially supported on the Web Server. on the web skus in add Item you should only have access to create a server based database. on express since it is a dedicated sku CE isn't an option in either the data tab or the add item. on vs you can create a CE database from the Data tab. and you should have access to create a server database in add item.
sql server management studio doesn't care where the source is. so it will work with all and create the database.
I use all the options listed just depending on what or where I am at the time. as was mentioned CE on the desktop is primarily a cache database perfect for offline database support. but does have its limitations and in addition really needs better integration. once I am created or working with a database in support I primarily use management studio.