Visual studio 2010 missing BI 2010 (SSDT 2010)

In my case, I had Visual Studio 2010 Enterprise on my system when I initiated the setup for SQL Server 2012 Standard. After the SQL 2012 got underway, it stopped and prompted for the VS 2010 setup.msi. At that point, I ejected the SQL Server disc and inserted the VS 2010 setup disc. Then I used the dialog presented by the SQL Server 2012 setup program to browse to the root of the VS 2010 DVD, selected the msi there, and opened it. The SQL 2012 setup continued and when I saw that it was again installing SQL components (rather than VS 2010 components), I ejected the VS 2010 disc (which caused the SQL 2012 setup program to display an error that there was no disc in the drive), placed the SQL 2012 setup disc back into the drive, then clicked the Try Again button. At that point, SQL Server 2012 resumed and after some time it completed successfully.

That is correct. Forward compatibility is guaranteed. Backwards compatibility does not exist^*. Having 4 different BIDS/SSDT instances on a machine is a pain I've known all too well but there's not much you can do.

There are ways of constructing SSIS packages such that your solution can target whatever version of SQL Server will be hosting them but that's installing, purchasing and learning a separate product (Mist).