For the features and functions you want, including "optical quality" and "good quality" as you have stated, I think you are asking too much for a max $350 price. For what you want, a <= $350 will likely be a huge disappointment with dark cloudy fuzzy glass, and likely to self-destruct with normal use, or have major consistency in POI issues, and faulty ability to track consistently when using turrets. I would expect at that max price the turrets will be mushy and fail in repeatability.
For hunting, you might be able to find good quality for "set and forget" capped turrets (which are fine for hunting) where you learn your holds high/low for distance, and no parallax adjustment, for $500 to $1000. I use one of these types of scopes (a Leupold 3-9x rimfire scope with duplex reticle) for grouse hunting, and it works fine with bright glass. However with the fixed parallax, I have to compromise focus. It is not a crisp focus at all distances that I shoot grouse. Even worse, for close up focus on the grouse for a shot say at 10-20 yards, I need to turn the eyepiece to have a slightly fuzzy reticle. With the duplex reticle, I work out my holds based on the grouse body proportions. But it still costs more than $350. For hunting you do not want dark cloudy fuzzy glass when peering for a head shot in the dark forest understory. You want high quality, light gathering, crisp focus glass with a heavy dark reticle, and that you have to pay for. Add the side parallax focus which is really beneficial, and you have to pay for that.
But when you add the target shooting purpose of use, I think you are needing the better quality glass and turrets closer to > $1000 and more if you are thinking about competition. You want a thinner reticle for target shooting than for hunting. If just plinking targets for fun, then yes you can get by for < $1000.
It is tough to find what you are looking for, for rimfire, that meets both hunting and target purposes. One thing to consider if using one rifle, is to mount a picatinny rail on the receiver for easy swapping out of two scopes with picatinny rings: buy one scope for hunting, one for target. If the rings and rail are good quality, the return to zero is fairly close.
I know, I know, I just inflated your price cap by about 5-10 times!
