Pretty much any "Youth" model out there will work; some examples of a few favorites:
Franchi 48 AL "Youth", 24" bbl, 20 or 28 ga (lotta other youth shotguns too offered in 20 ga from Mossberg, etc.)
CZ 452 Scout, .22lr
M1 Carbine or clone, .30 carbine (in fact, almost all military weapons have short LOPs, since they use iron sights and have to cater to the least common denominator in size of soldier)
Taurus 62 pump .22, Youth model (no longer made, but can be found used)
Yup, adjustable-stock AR15, any number of calibers.
Many makers offer their centerfire turnbolts in "Youth" configs as well (Rem Seven Youth, Browning A-bolt Micro, etc.)
Lotta other Youth .22s as well.
Would a scout style scope help with the eye dominance issue?
I believe the answer is yes it does - it helps me anyway.