With all three of your other Qs closed and this one probably too broad also I suggest you visit the Help Center even though you have been a user on TSE for more than three years.
Do all towns have fuel?
I have no idea but can't see that it matters. In particular there is no precise definition of town. Even focusing on Australia there is still imprecision towns or "urban centre localities" are commonly understood to be those centers of population not formally declared to be cities and having a population in excess of about 200 people. However I would imagine that fuel is available at least moderately close by wherever there are centres of population in Australia of more than 200 people because it would be impractical to travel long distances (burning fuel) to tank up.
are there likely to be sections longer than 250 kms (my approximate fuel range) without fuel?
I think not. The following seems viable:
~163 km BP Mt Garnet a great stop in the middle of nowhere
~215 km Ampol Georgetown Roadhouse
~217km BP Blackbull Worst service station. No customer service skills. Over priced. Gives you a rude mouthful of opinions when you didn't even ask for it. No customers or locals need to endure this pit stop!
~93 km BP Normanton
~190 km BP Burke & Wills Roadhouse, Four Ways
~209km Caltex Woolworths Cloncurry
~121 km Mt Isa
but since not all are open 24 hours what actually best suits you may depend upon your schedule. And for that please note WANTA.
The leg to Normanton is short because I have doubts about the availability of fuel on Route 83 after that point.
I would image it would be very effective, with an overall plan, merely to ask at each petrol station you stop at for advice about your next proposed stop.
What would be the required fuel range to complete the journey?
Enough fuel for about 1,200 km if trying to complete the journey without refuelling. However I would recommend allowing a significant margin of safety as a friend of a friend of mine did die in Australia from the consequences of being stranded. You may also want to carry at least a small reserve tank.