Assuming that both flights are on a single ticket, your checked baggage will be tagged LHR-ORD-SLC. After you clear both Immigration and Customs at ORD, you will drop your through-tagged baggage at the re-check desk just past Customs, and then go through TSA security to return airside for your departing flight; there should be no need for you to go to the AA ticket counter.

If a delay on your BA flight causes you to mis-connect at ORD, then BA is responsible to re-book you on another flight -- whether on AA, DL, or UA. Re-booking on AA would be the most likely scenario, as long as there is a later AA flight with availability.

Can't answer your question about BAH, but I'm sure that someone will be along presently to do so.