Answering just part (1), there do exist techniques to operate directly on compressed data without first decompressing it. Several specialized compression algorithms have been specifically designed to support directly operating on the compressed data.
The RoaringBitmap, like several other compressed bitmap index algorithms,
can implement intersection, union, and difference (AND, OR, and AND-NOT) directly on the compressed data without decompressing.
(In many cases, the result is found faster than running the same operation on raw uncompressed bitmaps, which is of course faster than decompressing into a raw uncompressed bitmap and then running that same operation).
(Some proposed applications of homomorphic encryption only require these 3 operators).
several text compression schemes support searching compressed text for words and phrases; this is often faster than searching the raw uncompressed text directly.
A few kinds of processing can be performed directly on JPEG images, MPEG video, and MPEG audio, without completely decompressing the data (operating directly on the DCT transform coefficients).
"A Survey of Compressed Domain Processing Techniques"
seems to imply that includes simple arithmetic addition and multiplication operations.