The freewheel wobbles because the mounting threads on the hub are not perfectly co-axial with the hub. The wobble is harmless.

Freewheel hub axles break because providing clearance to mount a freewheel requires the drive side bearing to be offset inboard. This results in a length of unsupported axle between the bearing and the frame. The longer this unsupported length (i.e., the more sprockets there are on the rear cluster), the longer the lever arm on this unsupported section of axle, and the more susceptible it will be to breakage. That's why axle breakage is rare on wheels with 5-sprocket clusters, and why modern, high sprocket count wheels use a freehub system that allows the drive side bearing to sit closer to the dropout than is possible with a freewheel hub.