Hi Rasmus
Jetstar is one of Australia's budget airline carriers and furthermore it is the budget equivalent to Qantas, Australia's national carrier.
with Jetstar the price of ticket only covers the flight nothing more everything else will come with a price , meals , headsets etc etc
i dont think the middle name thing will be a problem
Jetsat should carry on your checked in luggage to singapore like most other carriers do with connecting flights so there shouldnt be any problems there
if in doubt you could always just check their website
Jetstar