Coccidiosis is a protozoa that lives in the intestines of poultry (and in soil). Chickens as they get older and exposed to their poop and soil build resistance/immunity. The only time Cocci is a problem is when there is a "break out" or "overload" in the gut, this is what makes the chicken sick. This is where treating with Coccidiostats come into play - Amprol/Amprolium (Corid) is a Thiamine (B1) blocker that helps
slow the growth -it doesn't kill, it starves cocci of B1 to control overgrowth- until the chicken can start building resistance.
I don't know a thing about barnlime, except is can be used in recipes for whitewashing coops. Even if does kill Cocci(??) you will still have it on some level because chickens poop-Cocci is always going to be in the poop.
FWIW- This is my opinion - clean out the brooder the best you can and put your new chicks in there. Chicks need to be exposed to what they will be encountering in their adult life. Feed them medicated chick starter if you wish. ** For my chicks, I
intentionally gather soil from my chicken run (sifted, but it would have the adult poop in it) and place it in the brooder for chicks to scratch through and ingest, they are also exposed to their own poop (of course) and to the soil/poop they will encounter outside (this is done first week of life). This way they can start building resistance. I keep Corid on hand to treat if they start to have any symptoms.
Here's some reading to get you started on learning about Cocci:
http://www.merckvetmanual.com/poultry/coccidiosis/overview-of-coccidiosis-in-poultry
http://www.thepoultrysite.com/articles/1138/coccidiosis-control/