To be honest, if you start out thinking that the trip will be awful, it probably will be!
Australia is an amazing place (and I've only seen a tiny, tiny bit of it), and totally worth the journey - especially if it's to see your DC.
I understand your fear of flying, and also your concern about skin cancer, but both of these can be dealt with.
If you have the time, many airlines run courses for nervous fliers, so it would be really helpful to consider one of those.
Re skin cancer, Australians are incredibly well aware and educated about the risks of skin cancer, so you'll find that most places you go (beaches excepted), there will be shade provided. Hats, long sleeves and sun screen are all essentials (which would be no different if you went to a hot European country, either!)
Also, don't go when it's summer in Australia. Although even their winters are warmer than ours, you won't have the scorching heat/sun that they can get during the summer.
Snakes, jellyfish, crocodiles and dengue fever etc.
The risk from animals/insects and disease is incredibly low, as PP have said - and you're catastrophising about those, I think.
Dealing with the flight:
Being cramped on an airplane - I can't sleep at the best of times and can get quite anxious in crowded places. A bed would be amazing but business class is £5000+ per seat!
If you can't afford anything more than economy seats, then make sure you have an aisle seat, so you can stretch your legs. If you get a flight leaving the UK in the morning, you don't need to worry about sleeping for the first leg of your journey - you just pass the time by watching the films, tv box sets, playing online games, reading, etc. Enjoy not being pestered by anyone from work! Food is usually pretty good on the Oz flight too, in my experience.
For the second leg of your journey, you'll be feeling naturally tired, so you may want to sleep - but if you can't, don't stress about it. Stay awake for as long as you can. You'll arrive in Australia in the early evening, so by the time you get to your destination, you can go to bed at a 'normal' time, and hopefully will sleep through. (Jet lag is worse coming back from Australia, so you should find that you don't suffer from it on the way there).
PP have suggested breaking the flight by having a day or so in Singapore or Bangkok or Dubai (all places that the airlines fly through, depending on the airline). Personally we've never done this, as we just want to get there - and a stopover like this just prolongs things. But it can help and, as has been said, there are some great airport hotels with pools etc.
The cost - DH thinks there is no point going for less than 3 weeks. We usually go on holiday for a week in Europe so this would be eye-watering by comparison.
Your DH is right. There really is no point going all that way for just a week! And who wouldn't want more time with their DC? The cost shouldn't be eye-watering if you have time to plan, and don't have to be there on specific dates. If you can't stay with your DC, then there are some really good AirBnB deals - the last time we went to Sydney (admittedly a few years ago now because of Covid), our flat was £50 per night, so really good value. You don't have to stay in fancy, expensive hotels! (Obviously don't know what the availability of places would be where you're going).
I hope this helps - and that you can find a way to enjoy what should be a brilliant trip!