Only the "yellow stripe" Cali's like the Cerritos, the "red stripe" and "blue stripe" Cali's have different functions, possibly Diplomacy in the former case.
Yes, although I think the way the Starfleet divisions interact with each other is... complicated. If we take the colored divisions as being actual divisions, they all work together side-by-side cross command by default. The divisional colors signify both organizational division and role... so even though there are gold, red, and blue stripe California's, they're still in the same fleet division, the ships just have different roles. I.e. the red stripe Cali's would still be under the Operations Division fleet, the red stripe just signifying their role within it.
Given that Communications seems to the one of the main functions of Operations Officers distinct from from Engineering or Tactical Officers, my guess would be signals intelligence* as a specialist logistics officers would be a key part of many special forces units.
* The AR-558 mission was signals intelligence related though the team may not have been.
I can see that, although in the case of Starfleet it probably extends beyond specifically "Signals Intelligence" to be a still specialized, but more general communications outfit. I would imagine the station Barclay worked at, the "Communications Research Center" was probably attached to this.