In answer to this
Can I use the roll-ups to tie to another Parent table? I'd like to tie all these to the hospitals that call us. But in theory, it could one day be more than one hospital (e.g. patient transfers, two hospitals call us). I have a summary of hospitals, which today is universally one hospital deep -- but again, within the realm of possibility to be more one day.
It depends what you mean by tie
If you simply want to be able to view a hospital record and see an embedded report of all the cases that made reference to that hospital, then you would create a report link field on the hospital record and on the left side you would put the hospital name and then on the right side you would configure that multi select field or if it won't take the multi select field you would configure a ToText formula of the multi select Sélect field. BUT, then you set the Report link to accept non exact matches.
The effect of this should be at the hospital name is contained within the list of hospitals that was on the case record multi select field then it will show on the embedded report which is filtered by that report link field. It's not actually a relationship so you won't be able to roll up any summary totals like counts or things like that but at least it will show on the hospital record.