Culture spread does stack linearly, allegiance gains/losses do have a cap, however they depend on the culture difference between friendly/enemy. With no enemy culture, one of your broadcasts can max out allegiance gain (or enemy allegiance loss), but as the enemy builds up so must you to keep up the difference between the culture outputs.

They make culture spread faster, so in reality having more than one is almost always a good idea, especially on the border worlds. Culture spread drops off with each jump, so the more centers you have the farther it spreads in a reasonable amount of time