High Rasmus and welcome to the forum,

The absence of a middle name from flight booking should not be an issue and I do it most times when flying either international or interstate without a problem.
As long as Christian and Family Name are shown.

For the stop in Darwin, if you've booked all the way through to Singapore it could even be the same plane doing a refueling stop but even with it being two different planes it may depend also on Australian exit and if you're exiting at Darwin or just in transit and exiting at Brisbane.
That'll be determined on whether the Brisbane to Darwin flight uses Brisbane International or Domestic terminal - your flight information should tell you that.

If you do the exit at Darwin, you'll need to collect luggage, go through immigration and customs there and then check your luggage back in but it's all in the same terminal building at Darwin and a relatively small one at that.

Jetstar are the low cost subsidiary of Qantas, Australia's national airline and in that respect a well regarded airline on safety and all, though like all budget airlines there can be hiccups even without all that NH snow to be dealt with.

If it was later April you were arriving in Darwin, it would not have been too bad a time to make a stop-over of a few days, the wet season mostly ended but waterfalls in Litchfield National Park will be roaring as would those in Kakadu but still probably too wet to get into them there though the accessible wetlands would be magnificent with birdlife.
Still be very humid though with all that water about and higher temperatures.