ZPAQ is a defined standard format which embedds the decompression algorithm for each block as bytecode into the archive. One can define a transformation (eg. LZ77 or color transformations of a picture) and compress this data through context mixing and arithmetic coding. It uses checksums but I can't judge rightnow whether there is some issue implied concerning bitflips.

Yes, there are two examples for pi here: http://mattmahoney.net/dc/zpaqutil.html and one where the input does not exactly have to be pi, but can contain it: https://github.com/pothos/zpaqlpy/blob/master/test/mixedpi2....

But of course for real world use cases one would choose a general compression algorithm or write a specific one for a certain type of data you want to handle (because it will perform better than a general algorithm).