Do graphemes relate to communication and thought disorders? The question Does language and/or culture affect an individual's cognitive capacity? explores the possibility of  differences in cognition based on language and cultural variations.

In this question I am focusing on certain aspects of language only:
the orthography, in particular, the types of graphemes.

Syllabary graphemes are shallow orthographies, where each kana represents and maintains a consistent sound throughout that language (with some exceptions).
The well recognised scripts of some Asian cultures (Chinese, Japanese, Korean,...) use syllabary systems.

Alphabetic graphemes are more complex. 
English, and a good deal of European languages use alphabetic graphemes, where each letter and its usage represents a sound. The great variation in how one symbol (letter) can sound, depending on the context of the letter within a word or sentence, is referred to as a deep orthography. 

Given the level of complication with alphabetical languages;   I am interested if they
can  lead to increased incidence of faulty cognition, for instance, thought disorders, dyslexia, circumstantiality, clang association, phonemic paraphasia, blocking, neologisms,.. to name a few.

Have there been any studies on this, and if so, what are there findings?
We have this tendency to explore, in detail, language abilities and label them "disorders". Some are disorders, like those stemming from autism, but, in any traditional sense, none of the listed qualify. They are simply biases in what people are good and not good at or classifications of errors. There's a title of a talk by Raymond Klein I've always liked.

Klein, R. M. (2002, November) Dyslexia, dysdrivia and dystennisia: Is it reasonable to expect everyone to read well? Handbook of Cognitive Neuroscience chapter presentations. Lake Tahoe, Nevada.

Imagine if we considered weak tennis playing a social problem. We would carefully examine these poor souls and work out very carefully the defects in their minds that cause them to be outcasts of society. We'd label said defects and work diligently to rehabilitate who we could...etc...you can see where it's going. What we've done with dyslexia is similar. 

Of course there will be people who are good and bad readers and writers, and there will be characteristics that some of those bad (and good) readers have in common. And we can categorize them. This does not, in itself, a disorder make.

In any field of endeavour, if you change the requirements of something that's an acquired skill, which reading and writing are, then you change the set of cognitive abilities that facilitate that skill. Different people have different cognitive aptitudes therefore their abilities in different kinds of writing systems, and the translation of those to phonemes, will differ. Furthermore, some systems will be easier than others and in different ways. There may be systematic variations in how those errors work out.

If you had asked whether clay tennis players have differences from grass players would it really be a question worth considering? It's only one because there's some belief that the common errors in reading are named and the people who aren't good at it are labelled.

But, in spite of the fact that some people do it, I think it's pretty difficult to call any of those differences disorders.

(As an aside, several of the listed "thought" disorders are pretty independent of orthography. Most would be considered thought errors.)