You need to use which. And you need a comma, otherwise the which may be interpreted as introducing a defining or restrictive relative clause with blackboard as its antecedent:
Bob writes on the blackboard, which causes a screeching noise.
Swan in Practical English Usage (p495) has a section in relatives with the title: 'which referring to a whole clause':
The example sentence
*It doesn't meet the blind's need who want to read.
is ungrammatical (that's what the asterisk indicates), because the relative clause who want to read has as antecedent the noun phrase the blind, but it follows a different noun, need, which is the head of the noun phrase the blind's need. So the relative clause, which ought to ...
When writing formal emails to clients, and I am asking them to contact us if they have any issues, I use either "me or John" OR "John or me".
How I personally differentiate between the two is by putting the preferred contact's name first. If I'd prefer the client to contact John, I would put his name first. As both are grammatically correct (per the above ...
It's "and me" in this case
You're right, they're wrong. In this case.
"This is my wife and me" - Correct
To see why, just expand the clause using "wife and me" into two clauses: one for "my wife" and one for "me":
correct This is my wife and this is me.
incorrect This is my wife and this is I.
Who says "this is I"? People who think that using ...