My crossfit athletes practice all movements (including Olympic lifts)in either weightlifting shoes or "regular" shoes (Nanos) because during a competitive WOD more often than not they have to use regular shoes. If the WOD has rope climbs, running, box jumps, rowing, for example a weighlifting shoe is sub-optimal and there is no time to change shoes during the WOD.
For a regular person it depends on the goal. If the main goal is Olympic lifting performance I'd do every "standing barbell lifts" (Olympic lifts, squats, deadlifts, pulls, overhead lifts, etc.) with weightlifting shoes since the goal will always to improve the actual Olympic lifts and you want to train in the same conditions.
Someone just training to get big and strong should probably wear flat shoes for deadlifts, at least if he use a powerlifting style of deadlift since elevated shoes are a disadvantage on that lift (many powerlifters deadlift without shoes).
Pretty much everything CAN be done with weightlifting shoes (except maybe rope climb) but it is no optimal. Loaded carries and prowler pushing are best done with "regular athletic shoes", same with running, jumping, bounding, etc.
And is it normal to bring two pair of shoes? Most of my crossfit athletes have a gym bag the size of a military bag and carry 3-5 different pair of shoes to the gym.