Billy,
I think the Cricket Carbine w/ the Aeon scope is a very good combo. As far as the caliber, there are others on this forum more experienced than I who could offer you better advise. Ted did a video on the comparison of "knock down power" of the .22 vs the .25 caliber. There was a big difference. If your shots are close, hunting/plinking around the house with the .177 would do great. However, as you become more proficient with your shot and your game becomes more weary you will want to take longer shots. On another one of Ted's videos he stated the pigeons would bolt if he came closer than 75 yards. There are many U tube posts of guys taking game routinely at 75 yards and beyond. If you want to be competitive in bechrest you may have to consider a .25 or .30 caliber.
The .117 is like the 410 of shotguns. Some guys like the challenge of shooting a smaller gun and don't mind passing up shots that a larger gun could hit. Most guys that shoot a 410 are amazing shots! I personally went with the .22 caliber because I wanted an all around gun. This caliber should easily be able to take a squirrel or bird at 50 yards by an average shooter. If I get bold enough, I could get my feet wet in bench rest with the .22. Take your time before you buy and enjoy the process. Make sure you know what you will do with the gun but don't rule out what you might do with it as well. Just my thoughts.