In theory, Employees are prohibited from working for other companies.
What role does phrase, In theory, play in this structure? Is it any kind of modifier? If yes, what is it modifying?
Yes, it is a modifier of the main clause. It basically modifies the verb "are prohibited" because it is really that prohibition that is theoretical. It can also be thought of as modifying the entire clause: the answer to the question "what is theoretical/in theory?" is that employees are prohibited from working for other companies.
Or can we just say that this is the case of split predicate and equivalent to Employees are prohibited from working for other companies in theory.
The problem I see with the placement of "in theory" at the end is that it inadvertently modifies "companies." Whenever you see a string of modifiers (often prepositional phrases), you should be suspicious that some weird meaning issues will crop up. This sentence has three: 1. from working, 2. for other companies, 3. in theory.
When is comma used for phrase or clause? Can we just use it for a short pause or phrase linked to comma should always modify something?
Comma usage is controversial. The GMAT will typically test commas against other punctuation such as semi-colons, where there is a clear rule about the purpose of one vs. the other. The GMAT is much less likely to test a comma against no comma.
Outside of the GMAT, I think your sentence could either have the comma or not. It would really depend on the writer's preference.
With: In theory, employees are prohibited from working for other companies.
Without: Employees are prohibited in theory from working for other companies.
As you suggest, the comma often indicates a short pause when the sentence is spoken. As I wrote the above, I realized that "In theory employees are..." with no comma meant no pause and thus "sounded funny." It sounded less weird (to me) to skip the pause when saying "...are prohibited in theory..." so that was the only reason I moved the phrase. The subjectivity of this issue is exactly why the GMAT would steer clear.