"Speaking off message" is a popular idiom....
Answering the banner question:'Is “speaking off” a popular idiom?' first; the OED defines idiom as:
A group of words established by usage as having a meaning not deducible from those of the individual words
A form of expression natural to a language, person, or group of people.
"he had a feeling for phrase and idiom"
The dialect of a people or part of a country.
~ the important part of idiom is that it's about a use of language that develops within a community.
As RaceYouAnytime says "Speaking off message" is a popular idiom in political journalism, and became common usage in this area in the early '90s. Since then (in my experience) it has become a well-used & commonly understood idiom in other contexts such as business and public service instutions - at least in American English and British English.
Its use presupposes (1) that there is a common stance that a particular group (e.g.. management, party officials, employees, teachers within a school) should be taking (2) when reporting to another group (eg. employees, the press, customers, parents, respectively) and often that (3) there is some coercive power relationship which discourages expressing opinions which deviate from the "party line" (a much older political idiom). Its usage seems to coincide with the rise in influence of political managers and 'spin doctors'*.
As such it would be an entirely odd & unidiomatic to use it referring to something said within group of friends, where differences in opinion are usually not managed!
*see 'The Thick of It'(2000's) with 'Yes Minister'(1980's) for a good satirical comparison of the new era with old in British politics
To better understand "off message" (oddgirlout discusses this too) and thereby to answer your other questions, it might be useful to imagine a hyphen in the phrase: "Speaking off-message" (as opposed to "Speaking on-message"). The use in (British & American) English has become widespread in the past (25?) years:
"I going off-line for a few days"
"our discussion has gone a bit off-topic in the last 5 minutes"
"you're looking a little off-colour today" (feeling ill)
(the hyphens are not required but would be understood and accepted)
The line ‘won’t suffer idiots and fools’ refers to "suffer fools gladly" though quite why Kelly thinks it clever to add the superfluous "idiots" is beyond me!