items on the current sprint backlog should be approached in priority order, and one item at a time by the whole team.
I don't know who says this, I at least don't remember having heard or read anything like the emphasized text so far. Of course, it depends also on whether an item for you is a story or a single task.
If it's a story (usually consisting of several tasks), there might be a chance of achieving this. However, as you say, sometimes the story just doesn't include enough tasks to keep everyone busy. Also often the tasks related to a story strongly depend on each other, e.g. there might be a design session (involving part or whole of team), then one or more coding tasks, then code review, functional testing, documentation etc. Obviously one can't do functional testing before the coding, and so on.
Since everyone has to do something, there will be at least as many tasks open at any given time as there are team members (or pairs). Taking into account that sometimes tasks are on hold for various reasons (inter-task dependencies, information needed from external parties etc.), usually even more.
In one Scrum project with a team of 4 developers, we had a very similar situation. We did strive to take stories in priority order as much as possible, and usually we had multiple stories and several tasks open at any time. In the beginning we often had problems with several half-finished stories at the end of the sprint. So we realized it is important to keep the number of open tasks / stories to a minimum, i.e. always attempt to finish open tasks /stories first before starting a new one. But practically, that minimum was never meant to be 1.
As for the number of stories per sprint, we just put in as many as we could comfortably fit in based on our (story, then task level) estimations. That was of course greatly influenced by our velocity, which in the beginning was estimated too high. After a couple of months we chipped it down to 60%, and that value seemed to work for us.