People can't provide full facts if you don't provide full information
If the mother is claiming housing benefit for the property whilst living elsewhere, that's housing benefit fraud, not necessarily council house fraud and is a separate issue to whether the daughter has a right to live there.
Child is adult with family. Wife still pays the rent as daughter has never formally been made a joint tenant. This may need to be done shortly as the wife is likely to inherit and therefore be forced into the situation of renouncing the council house
Not necessarily. Mother could decide to live in the council house and do something else with the inherited house (sell or rent it or keep it as a second home), as long as she doesn't claim means tested benefits that would not be fraud. She's lived there so long she's almost certainly got a lifelong tenancy. If she does claim means tested benefits she'd be expected to either live in the council house, sell the inherited one and live off the money at a sensible rate (no buying a Ferrari or something) until it's low enough she can qualify for means tested benefits again. Or live in the inherited property, continue claiming means tested benefits and give up the council house tenancy. Again, this is separate to the other issues.
If the council property is solely in the mother's name she is breaking the terms of her tenancy by not living there and has effectively abandoned the property, meaning the council can very quickly take it back. Unless there's someone living there with succession rights to the property, which will depend on that person's situation both currently and historically. The council won't know the property is abandoned unless someone let's the council know or unless the people involved in the illegal occupation/abandonment accidentally drop themselves in it, which they sometimes do!
Who pays the rent isn't the council's business. The daughter could be legally living there and if her mother chooses to pay her rent for her, that's between them.