If I wanted to put parmesan cheese and garlic into sourdough bread dough (mixed into the dough, not just a filling), should I do it before or after first rising? Will it prevent a proper rise if I put it in with dough, water, and starter right away? 
If you are making sourdough you are working with natural yeasts in an acidic dough, so I wouldn't think adding parmesan and garlic is going to be a problem as long as you leave extra time for proofing. Parmesan is salty, and salt inhibits yeast. You are probably adding salt already so reducing the amount to compensate would make sense.   As for adding garlic in my experience it doesn't work that well just chopped up and in the bread whether you add it at the beginning or knead it in later. The flavor doesn't get out and you get chunks of intense garlic rather than a nice hum in every bite, although maybe that's the effect you want. I have a garlic and rosemary infused olive oil I use rather than adding garlic directly, another option is to put chopped garlic in olive oil (or butter, or any other oil) on very low heat for 10-15 minutes, this will infuse the oil with the garlic flavor which you add to the break. I haven't had any issues with this inhibiting yeast.   I would add these ingredients in at first mix rather than kneading it in later as kneading after first proof as you will knock out most of your air and damage the structure. You may need a longer proof but the structure will be better. Alternatively if your dough is stretchy enough (like a pizza dough or focaccia) you can stretch it flat after first rise, put ingredients in as a layer, then fold it over, then do your final rise and bake. That only works when you have a stretchy dough though.  