I can't imagine its anything different to a normal company car lease, where fuel, tyres, maintenance, fleet insurance, rego are all part of a package. Fleet insurance is pretty broad, typically as long as the company gives the ok for someone to drive it and they can do so legally, it is covered - with the normal clauses for deliberate damage, dangerous driving or illegal driving. If they are booked by the law for any reason, that's the problem of the driver in question, not the company.
Workplace laws are pretty clear that you are still on company time in transit to or from work, so from a liability perspective Rex is responsible OH&S wise (insert not a lawyer disclaimer, but has been made clear to me on several occasions in my employment after doing marathon shifts).
All in all, that is beside the point. The company provided them the car and a taxi alternative for travel. Maybe the policy was implemented badly, maybe it was never a good idea, it doesn't matter. I agree the investigation smelt like a witch hunt and that Rex stepped over the line in firing him. None of it is an excuse for the actions he was found to have done or accused of doing, particularly for someone in the position of Captain.