Hi Yen,
If you already have your plumbing in a slab, it's most likely to late to install the p traps without busting
concrete, except maybe on your sink. The p trap is essentially the first thing the water flows through once it leaves your sink or tub, and would be conected conveniently right below the drain. The sink is high
enough you could still install a p trap, but the bathtub is to low. Also your washing machine drain is a straight drop, with the p trap typically in the floor where it connects to your plumbing pipes on grade.
To put a p trap in any other locations, like outside, would defeat the purpose, as the function of a p trap is to provide a water barrier with enough resistance, that with proper venting, you don't get a back flow of air coming up out of your drain when other drains introduce water into your system. If your worried about bugs, you'll be best served using a stainless steel screen on your outlet.
The good news is with your open system, and being gray water only. You won't have much if any methane build up in your pipes, meaning it most likely won't create a serious hazard from not having the p traps in place.
This is for educational purposes only, and shouldn't replace the advice of building professionals or required building codes.
Hope that helps.