Details will obviously vary and i'm not an expert but i'll try to answer the question from what I understand.
web browser in an Australian household laptop, and a web server in a US data centre
Data from your laptop will pass through a series of diverse networks, it will first pass through your ISPs access network, then into their core network. Depending on their size and business model they may buy IP transit service from larger provider locally or they may buy their own capacity on an intercontinental route to get it to the US.
When it gets to the US it will likely be passed on to a major IP transit provider, and from there to the provider that hosts the server.
Providers will choose equipment based on their capacity needs. The networking world is increasingly trending towards Ethernet based technologies, but I'm sure there is a lot of legacy stuff about.
Are such backbones now exclusively fibre-optic?
Intercontinental internet data is indeed nearly always carried on fibre optical cables. Copper cables don't have the capacity and satellites usually have latency issues.
Roughly what diameter would this cable be?
Wikipedia claims about an inch. Very little of that is the actual fibers, most will be mechanical reinforcement and power conductors.
How many independent fibres would carry information?
Generally, because of the need for repeaters, long distance undersea cables have less fibers than terrestrial or short distance undersea cables. Your particular cable seems to have a total of 3 fiber pairs on the core of it's route. https://www.submarinenetworks.com/systems/australia-usa/hawaiki-cable
Does each fibre carry many individual channels of information, and if so how (e.g. different frequencies of light or some other mode)?
Yes, wavelength division multiplexing is used.
What rate of information is sent in a single channel
It keeps increasing, I have certainly heard of 100GBPs per lambda (wavelength) but more may well be possible now.
You hit a point of diminishing returns though, because once you modulate a signal it no longer has a single wavelength. So as you increase the capacity of each lambda you find you have to space them further apart.
and is it encoded simply by flashing on and off?
It used to be (and may still be on older cables), but nowadays more advanced modulation systems are used.
Is some form of active repeater needed at regular lengths
Yes, optical amplifiers are used. As I understand it there are two main types. Erbium-doped fiber amplifiers (EDFAs) which provide point amplification and Raman amplifiers which provide amplification in the infrastructure fiber leading up to the amplifier. As I understand it the two types are often used in conjunction.
Crucially optical amplifiers can amplify the wavelength division multiplexed signals without de-multiplexing and re-multiplexing them. However they do have a limited wave band, so there are still limits on capacity.
What hardware connects to the ends of the cable?
There will be equipment to split out the wavelength divison multiplexing, then special transciever units to convert that to or from formats that can be used to pass it on to other equipment.
I'm thinking no conventional router could handle Tbps
Routers have been getting bigger, Cisco advertise routers with a capacity as high as 260TBPs https://www.cisco.com/c/en/us/products/routers/8000-series-routers/index.html#~benefits
(as it would need to transfer thousands of bits per modern CPU cycle).
High end routers don't use CPUs to route.
Does classic IP routing ever really happen on a bulk scale,
Sometimes it does, but undersea cables are not usually owned directly by any individual ISP.
So the cable operator will sell wavelengths to ISPs that want them. The individual ISPs will then provide their own IP routing equipment of suitable capacity for their needs.
Typically an ISP will have it's own equipment at both ends of a long distance link. Interconnections between providers generally happen over local links within a city (or even within a datacenter).