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Not wishing to start a 'which is better' spat, but having spent several years cycling in both London and Paris, I don't think Paris is any 'worse' than London, I actually find Paris less stressful and with better cycle lanes than London. A couple of years in Sydney too, including commuting Bondi to Botany Bay by bike for a while. Agree for the weather, hot hot sun, and lots of hills. You might as well wear a helmet, during the day at least, just to keep the sun off your head. Keep hydrated.
I cycled around Sydney for a year or two without a helmet. I got pulled over a couple of times and only received friendly reminders so you should be fine, but as @WestcoastPete says, the laws are about to get a lot more punitive.
As for cycling in Sydney; it's not as good as London, but it's not as bad as, say, Paris. The car is king and there are roads that you just have to learn to avoid.
Also, traffic lights and laws work a bit differently, and you'll be taken by surprise when you appear to have a green light but pedestrians keep crossing the road that you're trying to turn into. They have right of way so try not to yell at them too much.
The other big difference is the weather. The humidity always seems to be above 70% so you'll struggle to get anywhere without sweating buckets.