Mix parmesan cheese & garlic into sourdough bread dough before or after first rising?
My initial reaction is that adding cheese and garlic could affect the rise, unless you are using a recipe calibrated for their presence.  Cheese would probably affect the texture of the bread, interfering with an even rise.  Garlic is fairly astringent and has some antimicrobial properties; this could very well mess with the health of your yeast.

If you're going to try adding these, it would probably be best to knead them in rather than trying to add them during the first rise.