Video / audio ground on ungrounded equipment
Relax. There really isn't a problem here.

Don't confuse signal ground with safety ground. For these signals, what matters is shared signal ground, and with RCA cables that's provided by the cable shielding. In fact there are good reasons for NOT having signal ground tied to safety ground.

Safety ground is generally not needed in nonportable electronic equipment for the same reason it isn't needed in small electrical appliances outside the bath or shower: they're generally safe by design unless you do something foolish like pouring water into them or running them with the box open. Hence, a two-wire plug is fine.

Similarly, "backward" connection is pretty much irrelevant. The AC is going to a transformer, and the tranformer's output runs the rest of the device. Transformers don't care which way the plug is flipped. the only difference that makes is whether turning off the switch leaves the transformer hot or not, and unless you have the box open that makes zero difference.

Everything is designed correctly. Your mother-in-law is safe.