That species is Corydoras habrosus. I have kept them, but not spawned them (though they were in a community tank so likely spawned but eggs were eaten by other fish). Their spawning will be near-identical to C. pygmaeus. The eggs I have seen are easily spotted, they are white and maybe 2 mm or 1/16 inch diameter, placed singly here and there. I have only seen eggs a couple times, so the fish obviously place them in hidden spots for the most part. I would not remove them for this species, as they are unlikely to eat them.
If you have the corys in their own tank with no other fish, I very much doubt they will eat the eggs or fry. Mine have not. The wood with Java Fern attached will be ideal to put in this tank. Some Java Moss on the wood should be OK. And floating plants are helpful to reduce the light. Forest fish do not appreciate overhead lighting.
Leaves can be most any from hardwood trees. I use oak, but beech, maple will work. In NA you may find it difficult to find any now, depending where you live, but some fish stores sell almond leaves which will work. Any leaf that is safe will, submerged, slowly decompose and this is what you want. Infusoria as Akasha explained will appear and fry will eat it. The liquifry is another method. The sponge filter another.
I wouldn't recommend a bare bottom though; a layer of fine smooth sand like play sand is better. This will encourage more food for the fry, and the substrate will be healthier than a bare bottom which will need to be kept very clean. I am only changing half the tank water once a week, same as always in every tank, and vacuuming the open sand areas during this.
Byron.