Vermin can be shot all year round, on your own land (get permission before wandering onto a farmer's field and opening fire!) but there are 2 different categories requiring 2 kinds of weapons:
The first can be shot with air rifles. These include: magpies, crows, rooks, jays, grey squirrel, brown rat, wood pigeon, feral pigeon, collared doves, rabbits, and mink.
Foxes are also vermin, but these you have to shoot with a proper rifle.
Then there is the shooting/hunting seasons for pheasant, grouse, coots, ducks, deer and other game animals. Check with your local CA rep or look it up online.
Dogs can be shot under the protection of livestock act, if they are worrying your sheep, cattle, deer, poulty etc and if you use a shotgun/rifle as with foxes and report the incident to police within 24 hours, as you would if you'd run over a dog.
You do not have to report it if you run over a cat, but for some reason cats don't come into the Livestock Protection act. Though my understanding is you are allowed to shoot wild ones at will as they come under "vermin". NOT pets; only feral or stray ones. The distinction can be hard to make, so I'd say leave well alone.