Assuming the gun is in good mechanical condition, .22 Short RF ammo. should not cause any problems. However, these are not accurate guns, as compared to modern .22 revolvers, so don't expect to win many matches. Stay away from .22LR rounds, of course. Ed. ( I purposely didn't mention .22 Long RF as you're not going to find any of that, anyway. )
Yes, assuming it is in shootable condition. All of the Ladysmiths were made in the smokeless era, and should handle standard velocity .22 Shorts. Just be sure to NEVER use .22 Long Rifle cartridges. I would limit its use. If you break something, it will be nearly impossible to replace it.