I would recommend a bolt action .17HMR so that you can get the full accuracy potential from the round. Also, having a scope is the best way to get the precision from the cartridge.
As far as rabbits, it will tear a decent hole in their stomach but it doesn't damage the meat to terribly. It is plenty accurate for head shots though. I have a 3/8" spread at 100yds all day long with my Savage Heavy Barrel .17HMR.
Lever guns are "fun" but when it comes to actual precision... I wouldn't recommend them. At least for head shots on squirrls. Lever guns would be just fine for coyotes and body shots on rabbits... but for the detail work, I prefer my bolt guns.
Also, Alexander Arms has a .17HMR AR platform that is superb. One of my friends has it and it is just as accurate as my bolt gun.
Regardless, I have no doubt that you will grow to love the cartridge as I have. I'm also a big fan of the .17WSM. It has become my favorite rimfire round. The ballistics at 200yds is the same as a .17hmr at 100yds.
If you want to stay with the .22 caliber, I would recommend the .22WMR also known as the.22mag. They have a little bit more ass end behind them, albeit you give up a little bit of speed. Hornady makes a 35gr round which is only a few hundred fps slower than the .17hmr, and you have double the weight behind it. Out to 100 yds, I would take the .22wmr any day over the .17hmr. Past 100 yds, is where the .17hmr really shines.
My largest Hog to date was taken with my .22mag at 40 yds. It was a headshot on a 400lb boar. He was quartering away, and i put the bullet right behind his ear into his brain pan. He dropped instantly.
That being said, the world is your Oyster when looking at those 3 rimfire cartridges. Also, Franklin Armory makes a nice .17WSM AR platform. Franklin makes a proprietary magazine for it, and as far as I know, they are the only manufactures of the .17WSM AR platform.
Take a look at the Savage B-Mag if you are wanting an inexpensive semi auto .17WSM rifle.