I live near cairns.
Flies are not bad at night.
Mosquitoes are a big problem in wet bushy areas at cairns,
You could run 10 km or so at the beachfront in cairns, the goldsborough to babinda golds field track is a good off road place for a long run with top swimming areas without crocs.
Walsch's pyramid is a great mountain run/ walk, up to 922m high in 3 km. Great views all the way. Most other mountains in the area are overgrown rainforest jungle
I would definitely have a run in the red centre, just stay on a dirt road that doesn't have too many intersections or main track and early in the morning,
"Park runs"are popular in Australia, every Saturday at 7am,