The easy way is to use add/remove programs.  But if that isn't working, I'd download a copy of SQL Express (it is free), and use that to remove the instance.

As a thought - does connecting as hdcvs-lims-sql\ss2016, 49559 work?  I expect that it will.

If so, then it is likely your firewall is blocking connection on port 1434...

When you say the original media files are not present, do you mean the installation files or do you mean the "c:\program files" folder for SQL express got deleted?
If...

I agree with Sue that MS will likely be your best bet.  But I know from talking with their licensing experts that they use the term "expert" loosly.  I talked...

As an alternate to using a cursor for each DB, could you use:
EXEC sp_Msforeachdb 'INSERT INTO #nonexistent (IndexName,TableName,DB)
SELECT ic.IndexName,ic.TableName,DB_NAME() AS DB
FROM MONITORING.dbo.IDXCHECK ic
LEFT OUTER...

I still think the most likely culprit is that you have 13K files to write to.  How many physical disks are you writing to?  I'd be willing to bet the...

It might not be a bad idea to turn this project of "add a column" into a full re-write of all of the software to utilize a better database design?...

Ah right.  If it runs from the command prompt, it works as you.
Is 'Homebrew01" an AD account or a local account on both machines?
If it is a local...

I'm guessing though that you have some method of connecting to the remote machine via remote desktop or have some way to manage that server apart from just FTP?

FTP will be a similar can of worms with a few fun new problems as you then have to have credentials stored in plain text.  Either that or have no...