Rabbits, squirrels ect you need a hunting licence and they have to be in season. Chances are it will be cold out which will change the point of impact. Rabbit population around here was decimated so I retired my beagles four years ago to let rabbits populate. Now they are fat pets.
Coyotes perhaps did them in from personal observations. So on to guns. I do not think either gun would be good for rabbits. So perhaps you need to rethink that part of it.
As for black birds and pigeons, rats either would do the job. Main thing is shooting the gun lots to get good with it. 177 has a wider range of pellets and less money to shoot than a .22. 22 would have more hitting power than a 177. Piston is better than a spring in winter weather. Spring guns are proven dependable over a lot more years as reliable.
Scopes the Ruger is ok for plinking pop cans but if you expect to hit a quarter inch dot at 20 yards every shot it is not likely going to happen. Perhaps spend that money on a better gun and learn open sights and save up more for a scope. Once you have a good scope they move from gun to gun which makes upgrading cheaper down the road. Nothing worse than not being able to see where pellet hits through your scope. But the scope is clear. Rings not much good and strip easy. Make sure you use the scope stop provided on the gun.
I have bought one terrific gun off trading post which was an Airhawk that had a lot of work done to the trigger and internals. I have three Airhawks and find them very good guns. But they are very large full size guns and if I went out to the bush all day I would take a different pellet gun because the weight of the Airhawk would kill my back. Since Airhawks went up $50 in price and no free shipping their are better guns for the money now. Airhawks have available parts if you want to change them and easy to change springs ect. .
I do not know the Classic so maybe others will chime in on that one.
Sweetest cheap spring gun I bought new was a Hatsan 135S form a store call Deals (Division of KW Surplus) no scope and gun was $170 and it has shot terrific since new and two years later still bang on. But comes with open sights, lighter, shoots good, little smaller in size and a great smooth safety. And no shipping charge which lowered price of the gun because it was store bought. I would shoot it up against anyone's springer.
I may have an Airhawk going back up in trading post in the next few days as I got my new bench gun almost ready. I own three Airhawks and are my main shooters of my bench.
Good luck with your decision. CHEERS!