While I'm sure others here could point out shows that fit the description better, Death Note has a complex mystery motif and its own cute kid detective in the form of Near in its second half.
Near's actually an adult, just with stunted growth.
No, Near is a teenager. When he's first introduced he's around 10 or something I believe, but after the time skip he's around 16. Then again, that's never explicitly said in the show...but Mellow was "almost 15 years old" before the time skip, and he's clearly older than Near.
Anyway, I strongly recommend Death Note if you haven't watched it. It basically fits what you're looking for and the English dub has very good voice acting. It's so damn good that I've spent a LOT of time trying to find an anime that's similar. So I'm excited that you mentioned this Case Closed show...I'll have to check it out now.
The anime I reviewed a while ago, The Mystic Archives of Dantalian , is a detective show. Although unlike Conan, it features a lot of magic and supernatural forces. The main characters spend most of the time solving murders. It hasn't been dubbed yet. I dunno if that suits you.