Most of the members are chosen based on their skill and motivation to perform the tasks, as by State meritocratic rules.
On everyday stuff, the Arcology Council makes sure the agricultural & industrial sections run smooth and logistics for supply & trade are in order. The Arcology produces a variety of high quality tradegoods in small supplies in order to finance itself partially.
They also watch over the Science department, making sure safety regulations are being followed and that their equipment remains up to date. This assures that orders for quality assurances and blueprint improvements for civilian use keep flowing in, giving the Arcology another source of income.
The Council also maintains the bureaucracy and papers necessary to keep several of it's programs and services flowing, like the Prisoner Lend-lease service and the Medicine-For-Vitoc program. While not the most exciting job, it is a very crucial task in the complex State to perform well.
They are also tasked with checking & approving all the immigration papers, verify none have malicious intent and keep the Welcome Program running.
On the PR side, each large faction has one or more representatives. This allows for easier negotiations and communications. Gallente by example have their Diplomat accompanied by reporters due to their emphasis to openness. Amarr on the other side hold great importance to the soul, hence their preacher is seen with their representatives. Minmatar on the other hand place great emphasis on the Tribes, so each major tribe has a presence, along with the official Republic Emissionary.
The Caldari compose the majority of the council however, as that is where the Arcology originated. The main population and supporters are Caldari so far. Due to this there is not really a specified diplomat or emissionary dealing with PR, but more a variety of councilmembers who act based on knowledge and skill in the required field.
The council is also tasked with resolving inner conflicts that sometimes arise, like the Pro-Heth and CEP-loyal groupclashes in the recent past, but also the new growing conflicts with the Minmatar versus the Gallente and Amarr populations. The aim is to solve issues diplomatically to prevent the use of Law Enforcements.
This doesn't always work, as currently the Gallente section is placed under increased Militia watch to protect them from increased aggression. Hopes are that the situation doesn't escalate further.
In short, the Council runs the Arcology and keeps me informed of it's progress and necessary adjustments. When needed I can overrule a decision, but this is a power I am reluctant to use.