Those shifters are different from the ones I have. Do they have a model number? Most Shimano 7/8 components are compatible, but there may be exceptions.
Could be a bent derailleur hanger.
Define "low sprocket" and "high sprocket". The largest sprocket gives the lowest gear ratio, so it's more clear if you state "big cog" and "small cog".
I begin derailleur adjustment with the shift cable detached. First step is to set the limit screws. Move the derailleur by hand from small to large cog. It should shift to large cog with light pressure, but not rattle or attempt to jump off the cog towards the spokes. Same at the small cog, should not hesitate shifting to small cog as you release pressure on RD, should not rattle. I will turn high limit screw in while shifting from 2nd to small cog until shift becomes impaired, then back it out 1/4 or 1/2 turn. Once limit screws are set correctly you can attach cable and set cable tension.