Post a photo of the chainrings in question. It takes a lot of wear to destroy a ring, or a set of rings.

Keep in mind that on new style multi-speed chainrings, the teeth will have a pulling side, and a back/slack side.

The back side of the teeth is often contoured to aid with shifting, and won't affect chain retention.

For the Single Speed, have you measured the chain length? It could be worn too (or a worn freewheel).

As mentioned, all freewheels tend to bounce around a bit when the wheel is spinning (coasting).

You probably won't notice a bent axle if the axle is fixed in the dropouts and the wheel is spinning.

If the rim is tight when the wheel is stopped, but wobbles when it is spinning, then that will be a spoke (truing) issue.

For better suggestions, try to explain exactly what you are doing. Pedalling, coasting. What parts are fixed (axle tightened into dropout), and what you are observing moving freewheel sprocket, rim, etc.