Nambroth, this is an excellent post! I live in a very isolated and rural setting and have used the crows as indicators of of raptor activity for years. I have even seen the crows diving at a bear that was too close to their nest. They can be loud, but in my opinion, that is a good thing. If the crows are screeching there's a very good chance there's something out there you might not want around. I have never had problems with crows in my gardens, bears, moose and deer, never crows.
I have never had crows mess around in my gardens either. Deer, rabbits, and bears are far worse. The only bird that does any damage in my garden are the ones that eat the sunflowers (seeds) that I grow for my chickens, but I don't mind it that much. Other birds in my garden tend to do far more pest control than harm.
I am not suggesting that crows aren't capable of causing garden damage-- they probably do, in some areas, but I just haven't ever experienced it.
Here is a good link to help us understand the relationships between crows and hawks, and crows and falcons.
About the first minute of this video shows a hawk being harassed by just two crows, notice the wing shape, wing diminutions, and the body size difference between the hawk and the falcon. Also look at how hawks want to stay close to the trees and hug the cover, this is because hawks can maneuver through and between branches and trees by virtue of their broad wings with large control surfaces. A hawks' wings generate a lot of lift at relatively slow speeds. In other words, by comparison to the falcon the hawk flies about as well as a hot air balloon. The two crows easily sends the hawk packing. The falcon on the other hand uses a fighter pilot tactic called trading altitude for speed, and speed for altitude.
The final part of the video shows a peregrine falcon harassing about 30 oops... better make that 29 crows.
This is a good point! Crows are not a good deterrent for falcons. Fortunately, falcon predation of chickens is very uncommon unless you live up north where gyrfalcons dwell.