Can St. Elmo's Fire on an aircraft cause instruments to fail or give erratic readings, primarily with glass cockpits? Are there backup instruments that would not be affected? Does it affect radio transmissions?
Yes, the strong magnetic fields which are created can cause all of those problems if it gets bad enough.
To quote the AIM where they talk about Precipitation Static (which is one method of creating St. Elmo's Fire):
b. A review of pilot reports often shows different symptoms with each problem that is encountered. The following list of problems is a summary of many pilot reports from many different aircraft. Each problem was caused by P-static:
1. Complete loss of VHF communications.
2. Erroneous magnetic compass readings (30 percent in error).
3. High pitched squeal on audio.
4. Motor boat sound on audio.
5. Loss of all avionics in clouds.
6. VLF navigation system inoperative most of the time.
7. Erratic instrument readouts.
8. Weak transmissions and poor receptivity of radios.
9. "St. Elmo's Fire" on windshield.
c. Each of these symptoms is caused by one general problem on the airframe. This problem is the inability of the accumulated charge to flow easily to the wing tips and tail of the airframe, and properly discharge to the airstream.