If the region that you're drawing isn't terribly large (you referenced an area about the size of Australia), it's reasonable to just map that region as a flat map and not worry too much about distortion. Any projection that you choose to declare it as will work because distortions are fairly small in most projections for a smallish area, especially if it's not a polar area (you need to pick certain projections for polar views). There are software tools that can take your regional flat map and reproject it onto another projection (equirectangular is a popular working format because it's very easy to apply to globes and other projections using readily-available software).

For a whole-world map, pick one that meets your intended use. If you have continental shapes and sizes that work well with interrupted Mollweide, then go with interrupted Mollweide (I think that Hammer and Hammer VII are prettier for whole-world maps, but it's a matter of opinion and the curved parallels make some folks unhappy).