I've been looking for any morbid anime or any scary anime in general and also puzzle solving anime I'm really interested in so if anyone could please help me find any anime in these genres. It would be greatly appreciated.

Spiral is an excellent one for puzzle solving:

http://en.wikipedia.org/wiki/Spiral:_The_Bonds_of_Reasoning

And although it's pretty old The Kindaichi Case Files is good too:

http://en.wikipedia.org/wiki/Kindaichi_Case_Files

For morbid trainwrecktastic goodness, Bokurano is a good bet:

http://en.wikipedia.org/wiki/Bokurano:_Ours

For dark and thought-provoking I'd suggest Shigofumi and also Ikigami ( I don't think there's an anime for Ikigami but there's a J-drama and a manga version ).

And there's definitely lots of puzzling in Liar Game which I think is also just J-drama and manga ^^

I'd suggest looking into Gosick and Un-Go as these are both good Mystery Series.

I also love Detective Conan but that's a very large series to get into but worth it. Especially if you use the wiki guides on the series and just stick to the episodes which arn't fillers.

Gore is hard as it's usually portrayed better in Manga then Anime.

If you don't mind live action gore I suggest everything and anything by the director Takashi Miike