Simple answer, yes you can reduce the size if the space is available. 
It shrinking advised? No. 
Refer to methods: https://support.microsoft.com/en-us/help/307487/how-to-shrink-the-tempdb-database-in-sql-server

Question though, why is your tempdb on the...

I don't see any code you developed to link any re-admissions to the original case nrs, but I'll get you started on determining the days between admission and discharge dates.

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This is based on current date, you'll need to replace 'getdate()' with the date you are working with'

SELECT datepart(year, CAST(DATEADD(dd, -1, DATEADD(yy, DATEDIFF(yy, 0, GETDATE()),...

We have a similar setup.
The one challenge so far was load balancing unexpected workloads on instances, because you are now sharing one resource pool between all Instances.
Other...

Question, why aren't you using the perfectly good SQL Maintenance plans to perform all of this?

That should create a new backup for DAILY, since you have the @fileDate portion in the filename. 
@fileDate is set to be a yyyymmdd format, so you'll be writing to the...

Yes.

Played around with it, can honestly not say I see an immediate use for it in my own life, but considering other similar products are doing this, it just makes...

Your INNER JOINS on Orders Orders ORD ON ORD.OrderID = ORDD.OrderID INNER JOIN rule out rows that don't match.
Consider a LEFT OUTER join to return unmatched orders...

To build trends in behavior over time, why your logspace is used and can't be reused, include checking log reuse reason with it maybe (suggestion)

select name,...