Question....

Was a train's crew considered a "team"...Meaning they pretty much stayed together and move from one assignment to another as a group..?? Or were they assigned to a particular engine, and went wherever that engine was needed, along with "their" caboose..??
And...If they were a "team", did they have a caboose of their "own", or did any ol' caboose that was available get put on the end of the train..??

The general process was that employees would bid on the jobs available in order of seniority. Of course, the characteristics of the ideal job could vary a bit, setting extra mileage against being at home each night. I think things would generally settle down for fairly long periods. There would be a time, possibly at timetable change, when the new job schedule would be posted and everyone would bid for their preferred job.
For a long time crews were allowed to adopt a caboose which would be assigned to their runs and which they would set up to their preference. Then management realized that they could squeeze more than one shift a day out of equipment and the cabooses were "pooled".
Similar status with locomotives where they were adopted to the point that an engineer could paint his name on the cab side. Mid 20th century, British drivers (top link) could have favourite loocmotives regularly, subject to their being available.