Get rid of the dashes in the date, they can cause issues.

UPDATE T_MYTABLE SET dateTransferred = '20180326 05.00.00' where id = 1223

In any contiguous 28-day period, there will always be exactly 20 non-weekend dates and 8 weekend dates.

You can directly search for "Backup" "failed" messages without having to pull the entire log file first, viz:

EXEC sys.xp_readerrorlog @ArchiveID, 1, N'Backup', N'failed'

When you create a weekly job schedule in SQL, you specify "Wednesday", as in every Wednesday (or specified identified Weds of the month), but you do not list all future...

No.  SQL Server itself does not store that type of info.  It would be too much overhead on the system.

If the table has a unique key, you could...

I strongly prefer to avoid code that requires a specific DATEFIRST setting; my code works the same under any setting.
But this:
WHERE DatePart(dw, d)=2
adds a dependency on the...

SELECT
    passed_date, DATEADD(DAY, -DATEDIFF(DAY, 0, day7) % 7, day7) AS calcd_date
FROM (
    VALUES(CAST('20180522' AS date)),('20180603'),('20180704')
) AS test_data(passed_date)
CROSS APPLY (
 ...

Why the need to over-engineer everything?  Each entry is, for example, "Wednesdays" not a specific Wednesday,  Simple enough.  Although I do agree that "Monday", "Monday" in the same row should not...

You're welcome, let me know how it goes.

Btw, it's inconsistent to use data type "datetime2" and GETDATE(), which returns only a "datetime" value.  That's why I used SYSDATETIME()...

UPDATE pr
SET fieldvalue = pe.requestind,
    datelstmod = SYSDATETIME()
FROM process pr
INNER JOIN person pe ON pe.personid = pr.personid AND pr.activityid = 991
WHERE...

This a common technique ("trick") to get a single line of values from multiple lines of input.  It's almost always at least a pair of conditions, e.g.:
MAX(CASE WHEN rline...