Definitely the Organ. One of my best friends is quite the prodigy when it comes to musical ability in general so I know how versatile these skills can be. Learning to play the piano would also be much easier in this case. My favorite music-playing character....I honestly don't have one. I mean the only anime with a focus on music that I have watched is probably "Your lie in April" and I have a lot of problems with the characters there. I can't think of anything else at the moment either...
Well I can't say why many people like it but I can certainly give you the reaon for why I personally like it so much. First of all I have to point out the stellar direction (I know nothing that tackles any of your points but worth mentioning). It plays with your expectations (see the café scene) along with those of Oreki the protagnoist. The art is simply beautiful and while the animation isn't excellent all the time when it needs to be good it can be really good. The soundtrack is quite varied and fits whenever it's used plus I'm a fan of both OPs and the second ending but that is obviously quite subjective. So much for all the "technical stuff". While it's not necessarily a mystery show in the classical sense its mysteries (which can be classified as such since they follow the formula of Question-> Clues -> Deduction -> Solution) in and of themselves aren't the main focus. They serve more as a developmental tool to develop/establish characters which is quite the interesting usage of a mystery in my opinion. In that ssense it's more of a slife of life character "study" than anythig else. Something which can be intriguing in its own right. As such it doesn't need an overly extensive focus on Drama or Comedy as that would only ruin the special atmosphere it has. It's a moderate show for sure (save for quite interesting and symbolic visuals at times such as whenever Oreki has his "hallucinations") but that for me is its strong suit. It's not over the top (although I can appreciate that as well), its characters compliment each other wonderfully, it has quite the good amount of messages regarding personal conflict, lifestyle choices and society (although the ladder one is debatable as its only really tackled through its effects on the character's psyche and personality). What can I say I really like the show.