You need 4 officers for 24/7 coverage; one spare, and one manager seems reasonable, iff having a PD is reasonable. Probably makes more sense to make use of the county sherrif rather than having a PD if there's not some issue with the sherrif.

City manager would be in charge of city business (building and occupancy permits, inspections, billing for services, etc), whereas the Mayor (and/or city council) makes the political decisions (setting the policies the manager executes, etc)

It seems a bit silly to be incorporated with less than a thousand people, but I don't know the local history. If there's not that much to do, the city manager is likely part time, and the mayor probably does mayoral stuff a couple times a month. (Quick monthly meeting + maybe show up to a community event or three)

The "city" structure is an administrative choice and it's super common down south to have tiny towns that run as a city.

A 6-person department is just about the minimum for 24/7 coverage. IDK why they have a department at all though. Just let the state police or sheriff deal with it IMO.