Any handgun that is not a revolver is a pistol- so, yes, there are .22 pistols. They include single shot top break pistols, derringers, and automatic pistols. The last category includes the makes of Ruger, Colt, Browning, High Standard, Smith & Wesson, and dozens of other makers. The link below is to a photo of a Browning Buckmark pistol- caliber .22 LR.
Ruger's website lists the current offerings.
S&w
Only if it's a .22 Long Rifle revolver. It's not compatible with the .22 Winchester Magnum Rimfire.
25-50 USD
You need to contact Ruger. You may have a prototype
Inexpensive, probably built pre-68
gatlin
High Standard 1959
It is a semi auto PISTOL, not a revolver. Value depends on exact model and condition. Roughly $100-$350.
The .22 Short, was developed for Smith and Wesson's first revolver , in 1857.
$100-$150
Best left to a gunsmith.