How can I show which data I'm using in my paper?
Both the problems you state are significant, and (I think) very problematic to the further progression of computer science as an actual science. I work in services / software engineering, and quite honestly, reproducibility of results is very, very bad in my work as well as in the work of others.

I will comment on both of your issues, starting with the second one:


  The data is either not at all available or it is not clear which version of the data was used


In many cases, this is actually the easier-to-solve issue. I would assume your department has IT resources (e.g., a department web space) that you can use for such purposes. Additionally, there are other, more field-specific, repositories that you can upload data to (for instance, there is the UCI machine learning data set repository or the Grid workload traces archive). Making data available to the public is really only an issue if you are not allowed to do so because of commercial interests of your data provider. For instance, I have access to real traces of executions of business processes of a big german logistics provider, which I am not allowed to give out. This clearly limits the usefulness of these traces to me.

The second problem


  The source code that was used to generate the results is not (publicly) available


is sort of tricky in practice. Clearly, throwing your code to Github is easy, but this hardly solves the issue of reproducibility. You will still need to include pretty detailed usage instructions and documentation to make this code of any use to another researcher. This may seem trivial for a naive 1000 Lines-of-Code implementation of an algorithm, but, for instance, my current research prototype approaches 25.000 lines of Java code plus a bit of XML and Groovy. Just putting this up on a repository is not enough, if we consider that a paper can only be considered reproducible if it is possible to do so in reasonable time. An additional problem in my field is that often prototypes are built for a specific execution environment. For instance, my department has a small OpenStack based private cloud, and many demonstrators that our students build realistically only run in this environment without drastic modifications.

I am currently in the transition between a postdoc and a more independent research position, and one of my personal goals is to make all work that I and my students publish easier to reproduce. However, so far we are only making baby steps, but at least we are trying :D