Most of those options are actually reasonably natural, but you might choose one or another if you wanted to emphasize certain aspects more.
The only one that sounds a little strange is #4 ("the body of Mike"). It just sounds a bit unnecessarily wordy (most people would just say "Mike's body" instead)
As for which to use when, it's mostly a stylistic thing:
"Mike's body" -- If you've already established that Mike is dead previously, then this is a pretty natural, matter-of-fact way to refer to it. It suggests that the fact Mike is dead is probably not the most important/notable aspect of the situation (so it can potentially imply a "run of the mill" feeling, i.e. that for the detective in question this is "just another body", and the fact that it's dead is nothing unusual to him).
"Mike's dead body" -- This emphasizes that Mike is, in fact, dead. It can give a feeling that the fact he's dead right now is fairly important in some way. (Perhaps the speaker knew Mike and is still somewhat coming to terms with the fact that he's suddenly dead, etc.)
"Mike's corpse" -- "corpse" is a more clinical or technical term, so it can give more of a sense of professional or scientific detachment. To the detective (or to the speaker), this isn't "eww, a dead body!" or "oh no, Mike's dead!" but instead it's "a dead body which belonged to someone named Mike, which is unfortunate but part of the job", etc.
"the dead body of Mike" / "the corpse of Mike" -- These have basically the same implications as their earlier equivalents, but since you're putting the body/corpse first, it focuses a bit more on the body, and less on Mike (the fact that the body belonged to Mike is something of a side-issue). These can potentially sound a little less natural unless there's some apparent reason why you're putting extra focus on the body as opposed to the person.
Note that all of these distinctions are fairly subtle, so any of the phrases could be used and it doesn't change things that much, though.