Jay, it may not be an intentional fishing trip, but it is a successful inadvertant one (and by my own informal rules, as I am responding, then you have another bite to add to the list, well done). Mispers are prioritised if there is perceived to be a threat to their (the missing persons), life. The first duty of the police officer is 'to save life'. I don't believe that 'car crime' is prioritised, but to use your own example, when some one 'does your garden shed over', they are likely to put the booty in a vehicle! At that point a car becomes involved, so would you classify that as car crime?
Regarding whether or not a helicopter would respond to a break in to a garden shed, all I can say (from my limited experience), is that if called by the force whilst the job is 'live', yes, there is a very good chance a helicopter would respond (depending of course on a number of variables). However, if you arrive home after work, to find that your shed was broken into some time in the previous eight hours, and the offenders are long gone, then I'm pretty sure that unless the offenders made off with a precious, life sized day-glo dinosaur, no.