First off, Welcome to IFI!
Annealing is for making the steel as soft as we can get it. Normalizing is for relieving stress, thereby minimizing distortion and/or cracking when we go to harden the thing. Note well: normalizing does not work for those steels with the most hardenability. Their stresses may only be relieved with a full anneal. (Normalizing is sometimes called "process annealing.")
For machining or other stock reduction, it is often better to spheroidize. Depends on the steel.
Grain growth is caused by too much time at temperature or too high a temperature. We re-nucleate new grain each time we heat the steel into austenite. This is why many smiths prefer to thermally cycle some parts. (They call it "normalizing three times.")
"Too many" heats for one steel will be a different range than for another steel. Decarburisation is something of a boogeyman. To hear some smiths talk, you can turn 5160 into 1018 by heating it once too often. For a better understanding of what decarburisation is all about, look up "Fick's Second Law of Diffusion." Yes, it is a problem (even a big problem) with some steels. But normalizing heats with oil-hardening or water-hardening steels will not see significant decarburisation.
I hope this is of some help.