Does gz compression ratio improve over time?

It does up to a certain point and this evens out. The compression algorithms have a restriction on the size of the blocks they look at (bzip2) and/or on the tables they keep with information on previous patterns (gzip).

In the case of gzip, once a table is full old entries get pushed out, and compression no further improves. Depending on the your compression quality factor (-0 to -9) and the repetitiveness of your input this filling up can of course can take a while and you might not notice.

Not much. The "distance" covered by the DEFLATE algorithm which gzip uses is limited to 32 KB.

Wikipedia link -> DEFLATE

It is worth benchmarking against the various gzip compression levels and also considering bzip2 and xz.