If a computer can handle compressed data with virtually no lag, then perhaps there can be a positive spillover effect to the network itself by the transmission of compressed data packages which are compressed/decompressed in realtime by the GPUs for near-zero lag. I think harnessing GPU power is definitely something worth exploring for future scaling, in more than one ways.
No. The data transmitted is mostly random and compressing random data has only a negligible benefit and even sometimes gets you the opposite of the desired result (compressed data ends up being larger).
I guess it depends on the nature of data.
If someone broadcasts a few bytes payment, I suspect you are right.
If someone broadcasts a 1mb/2mb/4mb solved block, that can probably be compressed as it will have redundant bytes.
If someone is sending the blockchain to someone who is lagging 1-2-3 years, well, he could send it in compressed batches. Surely that takes compression (?)
134212277 - blk00000.dat
99375871 - blk00000.dat.gz (74.04%)
97307825 - blk00000.dat.bz2 (72.5%)
...although the way it is transmitted might differ from the way it is eventually stored and gains may be less.