Weirdest call was my second day on field training when I was right out of the academy. I walked into a living room on a welfare check and there was 20 year old Indian guy, butt naked on the floor, covered in KY jelly, with his left fist completely inserted in his rectum screaming "come get me".
Weirdest item found would be a bed pan concealed down someones pants.
Those are great questions. I will answer your questions in the order you asked them. The first thing to remember with the detective position at almost all police departments is that it is a lateral promotion. In other words, you must start your career as a road patrol or beat cop, working the street taking calls. This is where you learn your investigative skills. At my department as with most, detective is a permanent position. At my agency now all of our detectives are "general assignment" meaning they could get anything from a burglary to a homicide assigned to them. My previous police department, which was about twice the size, gave detective specific assignments, such as sex crimes, violent crimes, persons crimes and fraud. If you become a detective you will be sent to advanced training in investigations and interviews. In my area, we can go take the same classes you would need to be a detective so that we can start building our resume before we have to apply for an opening.
The jurisdiction question is great. So yes, I work for an incorporated town within a county. My police "powers" are in full effect within the town limits, but I technically have police authority in the entire state. My agency has what are called MOU (memorandum's of understanding) with surrounding agencies. This means that the jurisdictions we have an agreement with allow us to enforce state statute in their territory. For instance, if I observe a traffic violation occur in my presence, but it is 100 yards away in the neighboring jurisdiction, the MOU allows me to drive over, pull the car over and lawfully write the citation without being legally challenged on my authority at that location. I have written tickets and even made arrests in other jurisdictions, depending on the circumstances. If I need to go arrest someone and it is more than a ten minute drive for me and my sergeant wont approve it, I have the local agency go pick the person up. This is because as LEO's in Florida the law allows what is called "transference of probable cause". This means I can call a cop anywhere in the state and tell them to go arrest a person in their jurisdiction, and they can do so based off my word alone that I have probable cause to arrest.