All the above is pretty good advice.
I took a 2nd spare which I didn't regret, as I took no tyre repair gear. I only got a flat on the Oodnadatta track on the way back. I took about 7 jerries of fuel as I only have a 60 lt tank. A roof rack was no drama, but keep the load as light as you can. Take warm clothes and togs for a swim in the hot springs.
It's a great trip. We did it east to west with ease. I have a front LSD but no locker. I did get stuck on top of a dune once, but there was an easier track I could have taken.

