Himeno

Would have been much easier to get QF7 and route AKL-SYD-DFW-DSM-DFW-BNE-AKL (and avoid the annoyance of LAX)

anyway...
Qantas will check the bags all the way through to DSM, but you'll have to collect them at LAX when you go through CBP.
You'll arrive at TBIT in LAX, go through CBP (and collect baggage), drop off the bags again just after customs, then head outside and walk to T4, go through TSA in T4.

DFW is a little harder since they won't know what gate the DFW-DSM flight will use when you depart LAX. You'll arrive in DFW at A, C or D and depart from B or D. They should tell you which gate shortly before landing. But DFW is all connected airside with the train, so no need to leave the secure area.