What language to write USB 3.0 application for ADC demo board?
The speed critical parts that handle real-time number crunching should be written in a compiled and efficient language, like C or C++.  1.6Gbps is 200MB/s, so depending on the amount of processing you're doing...let's just say you shouldn't write this in ruby or python.  

Now, if your processing is pipeline-able (doesn't require branching or moving up and down the program stack), or basically just a series of operations that are fixed and do not depend on the contents of the data, the same things will be done to all the data, then this is what things like OpenCL or CUDA were made for.  This lets you perform these operations on the computer's GPU, using it almost like DSP.  OpenCL works on all graphics cards, CUDA is Nvidia specific, so I would recommend OpenCL.  It also can fall back and use the CPU.

OpenCL is a framework rather than a language, but it does specify C99 as the preferred language to use with it.  

Also, Kevin Reid's comment is a great tip - SDRs do exactly this - process a flood of data coming from somewhere (usually USB 2.0 or 3.0) and an excellent place to start.  A simple one might be to look at how the rtl-sdr driver does it. The code is available here.