I will start this by saying I know absolutely NOTHING about the legalities of shooting crows with an airgun or discharging an airgun in your area. I'm going to answer based on what I know about crows and airguns but what I say shouldn't be considered to be advocating anything that's illegal where you live. It's up to you to verify what's legal and what's not.
A decent airgun should work at reasonable ranges. Something in 5mm (.20caliber) or .22caliber is probably a little better than .177 for crows. I've had good luck with a .20 cal shooting about 800fps.
You'll get 1, maybe 2 and then they'll leave. They might come back later, but it won't take long for them to learn that your area is dangerous and avoid it.
Conventional airguns using conventional pellets are generally safe, even in populated areas, if they are fired upwards at a fairly steep angle. Airgun pellets are very poor projectiles and are very light for their caliber since they are more or less hollow for much of their length. Therefore, after approximately 400 yards of travel they don't have enough energy to be considered dangerous and will have very low terminal velocity when they descend--although I wouldn't like to get hit in the eye with one even under those circumstances, I suppose. If you aim at shallow angles then there is a real risk of having a pellet miss and either damage something or injure someone downrange.
NONE of that applies to the very powerful pre-charged pneumatic airguns, especially when used with solid, bullet-like projectiles instead of conventional/traditional diabolo style pellets. They should be treated more like handguns or black-powder rifles in terms of how dangerous the projectiles will be downrange.