I usually use a liquid nail behind that skirt board and then nail it. The risers and treads should be tight, but dont force hem in too tight or it might create gaps on the previous step.
That corner, where the upper stair comes around the corner is going to give you trouble. There should be a small triangle shaped piece on the upper stringer so hat you can level off before you make the corner. On the lower section, I usually will come down plumb instead of making that 45 angle cut. You also need to give yourself more level surface wrapping That second corner. You just need more room to make your cuts.
I always corner bead first, if you don't, the mud might compromise your reveals