Just saw this for the first time it's been on my list for ages and my completely WAG is he killed the mom before he took the kids to the Outback. Since the girl is in the same kitchen as the mom was in at the beginning of the movie wouldn't she be there otherwise?

How can this be explained?
He was a sandwich short of a picnic basket.
Really what is their to explain?
It happens.

I think we're supposed to assume that his business/job had crashed in about him and he'd decided to end it all and take the children with him. As I remember the book starts with a plane crash and the children are the only survivors but I'm guessing that was just too simple for Nick Roeg.

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Weird that he takes his kids but not his wife

The boy was shouting 'the wheels's come off' - meaning his toy car - but I think this indicated the father's frame of mind.
"Someone has been tampering with Hank's memories."

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Remember that the father has placed a container of gasoline (or "petrol", as the Australians would say) in the trunk (the trunk being located in the front, since the car is a traditional Volkswagen Beetle; the engine is in the rear). (Not a safe thing to be carrying around in one's car, by the way.) His plan is that, if he should run out of fuel during the drive to the prospecting/picknicking expedition, or if during the prospecting/picknicking expedition he should decide to carry on with his life and return home with the children rather than pursuing the murder-suicide option, then he'll simply refill the car's fuel tank. On the other hand, if he should decide to pursue the murder-suicide option, then he'll pour the gasoline from the container onto the car and start a fire that'll render the car unusable by the children if they survive his attempts to shoot them.
Again, this is all way too complicated for an audience to fully understand on a first viewing when the information is presented the way Roeg presents it.

How can we explain the brief shot of the fuel gauge, shown just as the father starts the engine, showing zero? The placing of it suggests that the father knew it was virtually empty, so what was his plan?

The way I read the beginning of the film, the father is a geologist who believes he's located some potentially profitable mineral deposits in the Outback, and he's been making the rounds of potential investors in the city to try to get a mining operation going. He's had a particularly discouraging meeting with a potential investor; he returns home in the middle of the day in a despondent state of mind; he figures he'll take his two children on a short combined prospecting and picnicking expedition that afternoon, as a final desperate stab at productive activity and wholesome family life; but he brings along a gun, just in case he should decide that he can't take it any more and wants to kill the children and himself. Once the three of them are in the Outback, he chooses the latter plan, with the added touch of destroying the car so that if the children survive being shot they won't be able to use it to get to safety. In the end, though, his attempt to shoot the children is completely half-hearted and ineffectual. He does succeed in killing himself, though.
Frankly, this scenario, while interesting, is too complicated for an audience to figure out on a first viewing of the film when the requisite information is presented in Nicolas Roeg's fragmented, quick-cutting style. Roeg should have either gone with a simpler scenario, or eased up on his fragmented style to present the information more clearly.

His daughter was so smoking hot it drove him crazy.
This is the only reasonable explanation.

If the rule you followed brought you to this, of what use was the rule?

His daughter was so smoking hot it drove him crazy.