This is a question that will receive more response on the Number Crunching board. But here are a few ideas.

The relative resource shares between projects are used by BOINC Manager to determine which projects to request work from and which to let go idle for a period of time. So, on the projects tab, what is the resource share to Rosetta?

It may be me but I can't tell what the resource share is? How do I determine?
On the projects tab, does Rosetta indicate "won't get new tasks"? No, it just says Communication Deferred

In the messages shown when you do the update, does it indicate it is requesting new work for CPU? No just CD (see above)

Do you already have tasks that show they are running at high priority in the BOINC Manager? BOINC does not request work in that case since it already feels behind.

That is a pile of questions you bundled up there.

First things first, if your Rosetta task is not making progress, but BOINC shows it is "running", exit and restart BOINC.

When you switch between projects every 60 min. you are just telling BOINC to decide each hour which task would be best to work on. It may decide over and over that the same task needs work. Just depends on your resource shares, time to deadline, estimated time remaining, history of the amount of time per day your machine is actively running BOINC, etc. etc. In short, yes, it is figuring it can still run the Rosetta task on time, and so it is trying to rotate through your projects to match your resource shares. So it's not necessarily related to the task with the earliest deadline.