If the freewheel isn't seized, it sounds as though the chain is coming off a pulley to the side when you backpedal, pivoting the cage until enough force is applied to slip the chain back on. This usually happens with a misalignment of either the rear derailleur hanger or the derailleur/cage itself (seems likely in your case).

As for the shifting issues, we'd have to see a picture. The chain might be too short.