If I flew fast enough opposite the Earth's rotation would I go back in time?

Possibly , if you flew faster than the speed of light. The Earth's rotation would be irrelevant. You've watched too much Superman!

When we go from Australia to the US we arrive a few hours earlier than we left on the same day.

When we go back we arrive two days after we left.

Local time is affected by the motion but when you cross the dateline it all suddenly corrects itself.

You could only get to an earlier ( local) time going opposite the earths rotation if you stayed on the same side of the dateline.

Otherwise you suddenly LOSE 24 hours.

So an entire day ceases to exist.

I don't believe that's as possible as comic-fantasy lovers would like it to be. But if it were possible to fly in the first place, and to be able to defy gravity, breathe in space, and rotate the earth as you rotate to be able to turn back time, you'd have to be going faster than any machine ever made ever went. But I don't think one could ever stop, pause, reverse time...unfortunately. If you could do this (reverse rotation), the earth would be moving in the most unnatural sense or if it stopped even if it moved more quickly then ever possible and we could feel it, that could spell out catastrophe for the world.

Time travel backwards is impossible with our current knowledge. If you did somehow travel back only 2 hours, you would be 1 million miles away from the earth as the sun and the earth travel around the galaxy at 500,000 mph

Nope. Earth's rotation is just one of countless events that together define the passage of time. Understand...I talking about time, not the measurement of time which is an artifact of mankind.

No, that only worked for Superman when Lois Lane died.

But anything can happen in sci- fi.

The Earth is neither the center of the universe nor is it the center of time....

Time is continous and therefore it is impossible to go back, however, if you travel at the speed of light, your perception of time would be slower, and you would "go" to the future

Superman did

yup, Superman did just that many times so it is true.