As larger ships need more personnel to function fully, they usually have bridges with a number of console stations as opposed to a cockpit. The layout of the bridge and what specific stations will be on it will depend on the ship and its capabilities; the most famous of them would of course be the typical Star Destroyer’s bridge, featuring a central command walkway with two crew pits on either side.
Unfortunately, as I have just discovered after a surprising amount of digging, the different stations don’t seem to ever be properly defined, for a specific ship or as a general rule. I can tell you that a typical bridge will have communications, navigation and astrogation stations. On warships, the bridge will have stations for gunners. On ships with tractor beams, such as Star Destroyers, there will be people on the bridge able to operate them, possibly with the support of other people working nearer the tractor beams (it is unclear exactly how these are controlled). In Lost Stars there is a scene where an Imperial bridge officer gives navigational assistance to TIE fighters attempting to fly through an asteroid field, directing them to change course when needed; hence, there are officers capable of accessing tracking and sensor data.
Whether all these functions are carried out by different officers at separate stations is a bit unclear. According to the one diagram I managed to find of an Imperial bridge (from an old d20 sourcebook), weapons and communications are actually taken care of in separate areas, outside the crew pit. This doesn’t seem to be the case in new canon, but I believe that weapons, communications, navigation and astrogation would still have their own dedicated console stations, because they are more specialized and complex tasks. This specific subject isn’t well covered, so you’ve got plenty of freedom to interpret as you will.
As for names: generally, larger ships will get names, starfighters will not. Naming a ship is a sign of personalization, so names as opposed to designations are much more common among private spacers and organizations that allow a degree of attachment to the ships. The Rebels, for example, named a lot of their ships, and referred to them by that name, with the exception of the starfighters. The Imperials named only significant ships, ones where a name would add to the ship’s intimidation factor: capital ships, flagships, and ships used personally (often exclusively) by high-ranking officers. Small ships other than starfighters – gunships, shuttles, etc. – don’t usually get names. If a ship is owned personally by someone, it almost always gets a name; this means tramp freighters, yachts, re-purposed/stolen military vessels, and the like. Basically, just use your judgement; if the ship is unique and important, it probably needs a name.
In almost every case, if the ship has a name then the pilot/captain/crew will not have call signs. Call signs are usually given to the pilot of a one-person ship, typically a starfighter flying as part of a squadron. They are assigned based on the pilot’s squadron, and often follow a numerical pattern: Red Squadron might consist of the pilots Red One, Red Two, Red Three, Red Four, etc. Hence, every pilot gets a unique name that also identifies their squadron. This was common practice in the Alliance and New Republic, and though call signs also followed this pattern in the Grand Army it seems that the clone pilots often preferred to call each other by name. Imperial call signs followed very specific guidelines, so here is the article on that. If a communications officer needed to give instructions to a ship with multiple crewmembers (a capital ship, a freighter, a Star Destroyer, etc.) then that ship would usually be addressed by its name or designation, whatever is commonly accepted or known, and the captain/commanding officer would relay instructions to the relevant crewmembers.
I hope that helps! That’s a lot of information to get through, so feel free to ask for clarification on anything!
~ Jacen