To answer the questions about a young police dog after having trained a few:
They aren't really "fully trained" until they are mentally mature at about three years old, any time before that they are still a young dog. Even after complete training they require maintenance training on the order of several hours a week. While they can work the streets younger and do a good job this is when they mature.
As far as one being found wandering loose, I would suspect that it could be more likely than your average dog given the same circumstances. This is part of the reason you have to be so careful to secure them well. Its not that they would be any greater danger to the public (They should be confident and comfortable in strange situations to even be considered for a K9 program) but they are a greater danger to themselves. They tend to be fearless, fast, and have good problem solving abilities as well as be tenaciously curious. They are good at escaping enclosures and would then go wherever their curiosity takes them, and may not respect as dangerous situations that would frighten a normal dog.
Not at all.
The county and state police are completely separate entities responsible for different things. The county is responsible for more local laws generally keeping the peace while the state only cares for the main roads and larger organized criminal efforts unless contracted by the counties or communities to provide local enforcement or special services (SWAT, K9, labs, investigative etc.).
The Sheriff is always an elected official owing his job to the people, while the undersheriff who handles the more mundane administrative work is usually a career officer chosen by the sheriff. That way the Sheriff and local law enforcement is independent of the tides of local government and not beholden to the council or county administrator, that way they are free to investigate the government itself if needed. The police should ALWAYS be answerable to the public, they should never be under the direct control of anyone not elected by the people. This keeps them working for the welfare of the people not as a private armed force of some official.
The majority of law enforcement in the US is handled on a county by county basis, with only a little handled at the state level and properly even less at the federal level.
In general a good thing to remember about the US is that every government is independent of every other. While the county is below the state level there is no connection, or direction from one to the other, except where the state passes laws that effect they way counties can do business the same way they pass laws that effect how I do business. Its not like a big "Governmental corporation" With the federal office directing the state offices directing local operations, they really are separate entities.