To answer a few questions, there are several air rifles out there that are under lever or side cockers, you usually do however see them on higher end airguns, but they are out there.
For about the best (and they are VERY GOOD GUNS) check out flying dragon air rifles online and look at the Xisico series of guns. A better bang for the buck you can't really find.
Now I will caution you on airguns a little and get up on the soap box. FPS numbers sell air guns, and the numbers you see on the side of the boxes are usually with very light pellets that are not the best for the application you are going to be using it for. Second shooting a high powered spring air gun is really not for everyone, it is a very different mind set and you have to master special holds and such to put that little light pellet in a very small area to put the animal down in a humane manner. You are going to practice your socks off....good news is you can do this for pennies and in your basement or garage....but you are going to have to practice and eventually reach out to your "hunting distance"
As to the 177, 20, 22, 25.....on and on....well that is a topic that will be like politics everyone likes what they like for good reasons, you are going to have to decide on yourself what is best for you. I will say if you are going with big box stores you are going to find more 177 and 22 pellets and that is about it. And of those two you are going to find more 177 then anything else. It does shoot flatter, but has less mass, that means you have to be more accurate in your shot placement (practice again).
The move from a firearm to an air rifle is not as easy as many people think....some have no issue with the move, others come away saying that the gun is junk, the brand is junk after all they are sold at Walmart where everything is china and cheap junk. Bad guns do exist and slip past quality control of any company, but in reality most of the issues are with the shooter not using the correct hold and other issues....air guns can also be very pellet picky.
If I was to suggest a good 22 air rifle that will take smaller game/pests I would say go for a Benjamin 392....at about $170 it is a bit over your budget but you will have a rifle that will WILL last you a lifetime. They are also MSP, multi stroke pneumatic so you will have to deal with all that pumping, but you will have no spring recoil to deal with or learn the special holds, special scopes and such.
The 397 is the 177 version of this same rifle, and runs about the same cost.
Past these two unless your pests/game are birds that is about it IMHO for MSP guns, If you are doing birds you have the Crosman 2100 and that is a very good air rifle that is well under your budget. I currently have 2 of them and enjoy them very much, but much over a bird they are pretty iffy in the power area.
If you skip over all of this to the end (I do get wordy) come away with just this, you are going to have to practice, you are going to likely have issues when you start shooting powerful spring air guns. The double recoil is very real and will really send your shots all over the place.
If you have any other questions please ask.
Check this out on youtube.....he usually does very high end air rifles, but did one on his first rifle...watch the series....and check some other things out on his and other *good* channels.
https://www.youtube.com/watch?v=g3qHC7bzi-w