First off, do NOT get a semi-auto for your stated purpose! It won't cycle light rounds, litters the environment with shells that are hard to find, and isn't needed.
There are many many old rifles that will suit your needs, of the turnbolt, pump, lever, break open, and falling/rolling block varieties. Are you willing to buy used or new only?
I've owned many .22s in many flavors, and I don't have a single semi-auto left, for good reason.
One of my favorite .22s right now is a single shot *turnbolt* - Marlin 915 Youth. I love single shots, espec. single shot turnbolts. In fact, this is my home/ suburbia rifle, dedicated to backyard plinking with Aguila Super Colibris - deadly on squirrels, grackles, pigeons, and doves.
Now, if you want a repeater turnbolt, depending on budget, my advice, like that of many, is the standard "CZ, then Marlin or Savage". I prefer Marlin slightly over Savages. You'd have to go USED to find a CZ in your price range, but it can be done.
But I'm also a big fan of pumps and levers in .22s, because most of them WILL cycle the unusual rounds that you speak of .... whether it's a Marlin 39, Taurus/Rossi 62, or anything else, just to mention some of my favorites, you're good to go.
But also don't rule out a single shot break action, from Rossi to younameit. I love my T/C Hotshot Youth.... the backpack rifle, under 2.5 lbs.
If you MUST get a semi-auto, that leaves the Remington 552 speedmaster, because it's the only semi which will cycle shorts and such. My favorite semi-auto ever though, by far, was a plain Marlin 60 - and yes, I've had a 10/22, which was ok, but not as accurate as the Marlin 60.
And if you want a prettier gun with a wood stock and no plastic parts, you need to go used, or just get a CZ 452 or 455 and be done with it. The older the better - pre-plastic-trigger-guards, pre-plastic-stocks, etc.