Business Intelligence Development Studio installation for Visual studio 2010 problem
From your screenshot I judge that you are using the Express installer (It says: Select the express features to install), SSDT and Visual Studio 2010 have no influence on the options in this installer.

Make sure you install the Microsoft SQL Server2008 R2 RTM - Express with Advanced Services  as that is the only one including BIDS.

Have a look at Features Supported by the Editions of SQL Server 2008 R2 which clearly states that the regular express doesn't have BIDS and the Express with advanced services does have BIDS.

Installing BIDS has no prerequisite on Visual Studio or SSDT and SSDT has no influence whatsoever on the installer you are running.

Also, BIDS is by definition Visual Studio 2008 as explained here


  Business Intelligence Development Studio is Microsoft Visual Studio
  2008 with additional project types that are specific to SQL Server
  business intelligence.


If you're insisting on installing Visual Studio of a specific version you need to look into SQL Server data tools. SQL Server Data tools is not shown in the installation features but is a separate download and can not be reasonably used with SQL 2008