Yes. Make a map in the map designer where you use planet types that do spawn moons or asteroid (RandomTrue or the explict Terran/Desert etc. would work).

However, by default each homeworld has a "Starting Area" that includes asteroids and moons. To get rid of these, you'll need to save the map, then open it in Notepad and manually remove the asteroid and moon references from the starting area.

That is how you would do it, but I would not use that method. You will still need to edit each random map's starting area like I mentioned.

It's really not that hard to change. For every star that players spawn at, you just need to change this section of the map file.

playerParams
startingCredits 3000
startingMetal 800
startingCrystal 250
homePlanetType "TerranHome"
homePlanetStarRadiusRange
minPercentage 1.000000
maxPercentage 1.100000
areExtraPlanetsColonized FALSE
extraPlanetsMaxRadius 15.000000
extraPlanetsRadiusRange
minPercentage 0.900000
maxPercentage 1.000000
extraPlanetGroupCount 3
extraPlanetGroup
minCount 1
maxCount 1
planetTypeCount 1
planetType "Asteroid"
extraPlanetGroup
minCount 1
maxCount 1
planetTypeCount 1
planetType "Moon"
extraPlanetGroup
minCount 1
maxCount 1
planetTypeCount 1
planetType "RandomIceVolcanic"