This is my first time baking normal maps, and I am very new to the “high-poly -> low poly -> bake normals” workflow.
I finished the high-poly model, then went ahead to retopo it. Halfway through, I wanted to “test” the normal map baking and to my disappointment, it turned out like this:
You need to fix the normal of your high poly before baking, I think. The geometry can be more dense but can’t be defective or the defects will be baked into the map. Try remove doubles and recalculate normal on the high poly - be sure to select all verts before doing this or it’s no use.
I’ve recalculated the normals and removed all doubles, yet the errors persist 
If the low_poly model clips through the high poly model, will that ruin the normal map?
No it won’t ruin the normal map.
How does your UVMap for the low poly look? Is it properly unwrapped or did you use some automatic method (because it looks as if all the triangles are unwrapped separately)?
Is the low poly smooth shaded as it should be?
Try adding a triangulate modifier with the method other then automatic (like fixed) to the low poly before baking. And then use the same method to triangulate the low poly model you’ll be applying the normal map to.
