First, I already understand the basics of radio communication at this point, as that is all I've found searching elsewhere on the web for my question. (For example, How does NASA get information from space probes?)
What I really want to know is what a scientist might see at a workplace for tracking probes. Specifically, if something goes wrong with a major spacecraft, does someone see a literal red light start blinking? How can scientists find out when something goes wrong?
It's best to read a first-hand and also two second-hand accounts of a real red-light spacecraft emergency. SOHO, the Solar and Heliospheric Observatory was almost lost - more than once! It was - and still is, thankfully - in a halo orbit around the Sun-Earth Lagrange or Libration L1 point - about a million miles in front of the Earth towards the sun. This is roughly the same general neighborhood as DSCOVR, where they both keep an eye on the sun.
Briefly, SOHO's orbit was not stable, so it required regular station keeping maneuvers - short burns every few weeks or months's. It's design and orbit were configured so that it would slowly drift towards the Earth, and a sunward burn - perpendicular to its orbit, would put it back in place.
You can read a short, medium, or long version of the exciting story below. I strongly recommend it!
Saving Soho Leonard David, 1999
SOHO’s Recovery – An Unprecedented Success Story F.C. Vandenbussche, 1999
The SOHO Mission L1 Halo Orbit Recovery From the Attitude Control Anomalies of 1998 Craig E. Roberts, 2002
Roberts 2002 is the definitive report and review of the entire situation, problem, and recovery.
In short, you get a hold of some radio telescopes and listen carefully, analyze doppler shifts, and run computer simulations to compare and try to figure out what is happening.
You can see some more information about SOHO's orbit here.
This depends on the space probe, or more exactly, the design of the software that's used in the operations center. These days software like LabView is often used to convert the incoming data into a presentation. When data falls outside its normal values, the presentation can be modified, including blinking red lights if desired.
But this depends on how well you can define "data falls outside its normal values". When a tank pressure suddenly drops from $value to 0, that's a pretty good indication of a problem. But when the error is more subtle, you may not be able to use automated means to detect that there is a problem.
