This is specifically talking about public services within the duchy of Feann. Transport information can be found here.
It is going to be essential to have decent medical care of some description within the duchy otherwise disease could spread unchecked. Since there is no information on how medicine will work within Chronicles of Elyria, I'll keep this section brief and vague.
Every town should have at least one medical station that citizens can use to stay healthy. If people live in rural locations, then it might be possible to have travelling medical professionals visit them at certain times so that their medical needs are also met without having to travel into town. Having said that it might not be practical or affordable. In these situations, people living in rural areas will need to travel to the nearest town to get medical treatment.
In times of war, it is going to be essential to have medical staff at the front lines to deal with injuries caused by battle. The same is true on ships. At least one doctor will be required on a ship.
Regarding funding, it is going to be the responsibility of the Duke to fund medical care within the duchy, but it will probably be mayors and barons who will sort out medical care on the town or city level. This is because the Duke will be too busy with duchy level things to micro-manage each town. Having said that if there are any complaints about medical care in towns or cities, then the duke will step in to try and come up with a decent solution to the problem.
Law and order is going to be a very important aspect in Chronicles of Elyria. Without a doubt, there are going to be large player groups who will want to operate as criminals or try to bring down the duchy and thus some level of law enforcement is needed. What is not so sure is whether law and order should be handled by the military or whether it should be handled by a civilian police force. Having said that it would be a good idea to have some sort of presence in each town and a headquarters in the cities. That way if people have a problem they won't need to travel far to report problems to the local police department. Also, major roads should be patrolled by law enforcement to hopefully cut down on thieves or other types of criminal.
Funding will be handled by the Duke when it comes to law enforcement. In the same way as with medical services if there are any complaints about law enforcement then the Duke will get involved to hopefully sort out any problems in a satisfactory manner.
Since most of the buildings are most likely going to be made of wood (at least at the start) a fire service is going to be needed and quickly. As with the police and the medical service, this will be handled on the town or city level with the Duke financing the service.