I believe adding a car or hotel to an existing flight reservation does not change the booking into a BA Holiday. That is only when multiple items are booked at the same time as a package.

BA arrives at ORD T5, even though return flights depart from T3, so you will need to clear customs and drop your bag in T5 and then proceed land side to T3 for the AA departure to SLC. It's a bit of a pain tbh but should be easily doable in 3h30, allowing for some time in the (not bad) AA flagship lounge T3 before your SLC departure.