I'm a Cairns local and have been a pro diver here for many, many years! So first off Cairns is considered the gateway to the Great Barrier Reef and as a consequence, there are many options of trips you can take.
See
here for an explanation of the different areas of the Reef, and
here for overnight trips.
The Reef is making a comeback and is still a fantastic place to dive so don't believe all of the doom and gloom stories. The best areas are the Ribbon Reefs in the far north and the Coral Sea (30m / 100ft viz and deep waters with a lot of pelagics - shark feeds etc).
Mike Ball and
Spirit of Freedom are the 2 trips that run twice a week but book out months in advance so get in early if you want to dive the Ribbon Reefs.
The best time to dive the Ribbon Reef areas is June to September (our winter) when it's Minke Whale migration, so you get to dive with these creatures. Amazing experience.
From a weather point of view, the best 2 months are September and October when the seas are very calm, it's coming into summer but not humid yet. Warm water, calm seas, and really good viz. It's still off-peak tourist season so lots of availability and for the Outer Reef there are a lot of options.
Late November is the start of the monsoon (rainy season) so hot and humid, but still beautiful out at sea. getting pretty busy, and December is very busy with some boats charging Xmas surcharge of $100 around Xmas day etc.
Outer Reef is always nice and for all levels of diver. You can do 2 day 1 night, up to 4 or 5 days out at sea. Most do 3 day 2 night trips where you get around 11 dives. These trips run daily so will fit into most travel schedules.
Further down the coast the
Yongala runs out of Ava Beach (close to Ayr) and is an amazing wreck dive, but not the GBR so would be a dedicated trip just to dive the wreck.
We had the first cycle in around 5 years here a few weeks ago which was a fizz. It's the following rain and floods that wiped us out and that's rare.
Cyclones are not that common and again having lived here for over 30 years I can count the number of cycles affecting me directly in Cairns to be about 6. Low pressure systems in the Coral Sea can form rough conditions at this time of year, but it's rare trips cancel unless the winds get up to 30 knots (again not that common)
Feel free to drop me a DM if you want more info about the Far North or Cairns even if it's not diving related. Happy to help out.