You can include some mobility elements in your warm up, but not in place of warming up.
You’ll want to leave anything that hits deep for after your workout.
Start with addressing the hip.
Yes, you can do mobility following your workout or at a completely different time.
You can hit different things at different time throughout the day.
As you are working with an area you will see the benefits last longer. In the beginning the change may not last as long as you think it should. As you work with it and your body learns the new position/movement pattern it will stay longer. After things get worked out and the area isn’t as tight any longer it will be more maintenance and staying on top of it.