Will SSDT-BI work with SQL Server 2008 R2? Sorry if this is redundant, but due to the crazy naming of the tools, it's hard to find the answer to the question.

Question 1

Will SSIS packages, reports and so forth built with Microsoft SQL Server Data Tools - Business Intelligence (SSDT-BI) for Visual Studio 2013 work on SQL Server 2008 R2?

Question 2

I'm currently using SQL Server Business Intelligence Development Studio (BIDS) for Microsoft Visual Studio 2008. I want to potentially upgrade to Data Tools - Business Intelligence for Visual Studio 2013.

I assume I would need to A) purchase a new copy of Visual Studio 2013 and then B) download the free SSDT-BI software? That's assuming SSDT-BI for VS2013 works for 2008 R2.
Step A in Question 2 is not necessary.  The SSDT-BI package contains Visual Studio 2013 shell and will install it for you if it isn't there (see the additional information).

As for will it work?  Kinda.  You will be able to run packages which load data into SQL Server 2008 R2, but you will not be able to take advantage of the project deployment model released in SQL Server 2012.  

The reason why I answered Kinda is that you may have some problems running new packages using dtexec on your old server.  Microsoft's guidance on this isn't much more than a stopsign... So YMMY (and please post back here with your experience!)