Based on the stuff you don't like and the ones you are seeking, it seems as though we like similar animes and thus, I can give you some good suggestions:
Black Lagoon: The protagonist Rock is your average Japanese salary businessman but unfortunately, his collegues and superiors sort of betrayed him somehow and he ended up with a group of sea pirates, nothing to do with Somalia, of course. This anime already have two seasons and it will get a third one sometimes by the end of the year or the next.
Death Note: I don't really like any of the characters but Tsugumi Ohba is no doubt a genius and thus, you should give it a try. Or perhaps, you have already seen this.
Devil May Cry: This is the promotional anime series for the games and disappointing thing is that Dante will never use Devil Trigger with a slight possibility near the end though.
Higashi no Eden: This anime starts off like a rip-off of the Jason Bourne films but it changes into something totally different. Generally, the protagonist is definitely not Jason Bourne as that his goals are to protect and save the NEETs from warmongers and the ambitious bunch. Nonetheless, there are still multiple instances of mysteries and political conspiracies. The opening theme is done by Oasis which is pretty good. Its rare for an European group to do anime songs.
Kara no Kyoukai: This series of films of a varied lengths are very good and the fact that it is from TYPE-MOON and is part of the Naru-verse makes it slightly better. Not sure if you would consider this as strange stuff as that the female lead have a tendency of killing people.
Persona -trinity soul-: Like the series of games, this is not for kids. Aside from the fact that the story was somewhat confusing at the beginning and for the most part but you do get a reasonable explanation by the end of the series if you have the time and patience.
Phantom ~Requiem for the Phantom~: I know the protagonist in the original visual novel was a high schooler but the entire story had quite some major changes and thus, it fits your description perfectly.
Shikabane Hime: Unless you have the patience to watch both seasons, you might end up dropping it at the beginning like I almost did. For the first few episodes, I found the high schooler protagonist to be out of place. The guy is wandering around in the middle of the night getting involved with the undead for no reasons at all. Its like he is one of the braindead zombies. However, the series is very good after awhile. Not sure if you will feel the same way though. It is somewhat annoying at the beginning of the series when the guy acts that way.. Just to let you know though, you might find the female lead to be a rip-off of Dante from DMC for the first few episodes but that changes real quick afterwards. That was the reason why I started watching the anime in the first place, I thought Makina looked like a clone of Dante in the trailer but it changes later..
Speed Grapher: I have no clue what this is about as that I have never seen it but the poster looks somewhat cool. The least to say, the main characters are mostly adults and the story is nothing of a laughing matter. Probably some serious politics or scientific experiements of some sort. No idea!