Another vote for the Marlin X7, unbeatable in its price range. Much nicer, IMO than the Axis or the Ruger American. Also offers the option of a Boyd's laminated stock, if you don't care for the tupperware, and would probably still be under the $500 budget.
The T/C venture is another viable option, which I haven't tried but a friend in Montana has one in .300 Win Mag and loves it. I am a fan of the 5R rifled barrels, and have heard very good reports of the Venture's accuracy.
I love my Stevens 200 in .243, after my son sanded down the stock, installed a Hogue recoil pad and Rifle Basix trigger. It might be the most consistently accurate sporter rifle we have. (Too bad Savage discontinued it, I liked it a lot more than the Axis/Edge).
The Howa/Weatherby Vanguard is also a nice solid rifle, but noticeably heavier than the Marlin X7, and would push your budget a little.
I have owned and hunted with rifles with 20", 22" and 24" barrels and frankly don't see much difference between them. I certainly wouldn't exclude a rifle because it has a 22" barrel rather than a 20".