The summary.
PROPAGATION IS CACHES EXPIRING, NOT A SPREADING PROCESS
Which is why a change appears for some people and not others, and why records added where the nameservers do not point achieve nothing.
Check what a name resolves to, from more than one resolver.
A PORT IDENTIFIES A SERVICE ON A DEVICE
Testing whether a port is reachable distinguishes a blocked port from a failed service.
And a firewall rule for one transport protocol does not cover the other.
STATUS CODES SAY WHICH SIDE TO INVESTIGATE
Not found, forbidden, server error and service unavailable each point somewhere different.
FOR A CERTIFICATE WARNING, CHECK THE CLOCK
It is checked last by most people and is frequently the cause.
MOST REPORTS OF SLOW INTERNET ARE WIRELESS PROBLEMS
Test the same thing on a cable.
FOR MAIL, PREFER THE PROTOCOL THAT KEEPS MAIL ON THE SERVER
The other produces reports of mail disappearing from other devices.
Most client setup failures are the username not being the full address, or outgoing authentication not enabled.