This question is still a little broad...
On the issue of corpus callosum (CC) and mental illness, it's not clear that this follows the same pattern as ambidexterity. Highley et al. (1999) found that there fewer fibers in CC of women with schizophrenia:
A reduction in the total number of fibres in all regions of the corpus callosum except the rostrum was observed in female schizophrenic patients (P = 0.006; when controlling for brain weight, P = 0.053). [...] With age, all subregions of the corpus callosum except the rostrum showed a significant reduction in cross-sectional area (P = 0.018) and total fibre number (P = 0.002).
They also found "little change in the males".
In fact the press release you linked does not mention corpus callosum.
So whatever common factor drives ambidexterity and mental illness, it doesn't seem a simple as the (higher) number of fibers in CC.
Also, cross-dominance is a broader concept when it comes to the brain; the press release doesn't mention this term either. Even though Wikipedia considers it synonymous with mixed-handedness, one can have a cross-dominant vision but not cross-dominant handedness, for example. And in fact some of the more recent research has pointed out to different correlations with mental illness of these various types dominance, e.g.:
The rate of left or mixed-footedness, eye dominance, and any anomalous lateralization, but not handedness, discriminated between those who developed schizophrenia spectrum disorders and those who did not.
Of course they might not have had enough statistical power to detect it (for handedness.) But also a mini-review of prior work on this in the same paper points to somewhat contradictory conclusions, even for handedness, but still says that
The preponderance of evidence, however, supports the hypothesis that
schizophrenia patients are more often left or mixedhanded compared to controls. Left or mixed-handedness in schizophrenia may indicate abnormalities in
neurological structure or functioning.
Interestingly this issue of failing to detect the effects of dominance in handedness has been encountered before:
Several cohort studies have measured childhood
laterality as it relates to later risk for schizophrenia
(e.g., Crow et al., 1996; Cannon et al., 1997). Crow
et al. investigated handedness in the National Child
Development Survey, a study assessing all individuals
born in the United Kingdom during a single week
in 1958. Mothers of children who later developed
schizophrenia were more likely to rate their children
as ambidextrous at age 7 as compared to mothers of
children who did not develop schizophrenia. Additionally,
children who eventually developed schizophrenia
were less strongly right-handed than controls
at age 11.
Cannon et al. (1997) studied laterality in a total
population of 5362 children born in England, Scotland,
and Wales during one week in March 1946.
Results indicated that at age 11, children who later
developed schizophrenia showed a significant excess
of mixed-eye dominance as compared to the remainder
of the cohort. The authors failed, however, to
detect a significant difference in hand preference
between children who later developed schizophrenia
and children who did not. Nevertheless, Cannon et al.
interpreted their eye dominance findings as an indication
that lateralization might play a role in the
pathogenesis of schizophrenia.
And in the genomic age, some genetic candidates correlating with both lateralization and some mental illnesses have been identified...
Common polymorphisms in the gene PCSK6, whose protein product mediates the development of brain and body asymmetry through the NODAL pathway, have recently been associated with handedness in three studies, making it a key candidate gene for understanding the developmental and expression of human lateralization. We tested the hypothesis that the PCSK6 VNTR polymorphism rs1053972 influences the expression of handedness and aspects of dimensional schizotypy and autism. For a sample of 709 healthy individuals, rs1053972 genotype was significantly associated with categorical measures of handedness, and with dimensional handedness in subsets of the population with high schizotypy and magical ideation or a lack of strong right-handedness. Both findings showed evidence of stronger or exclusive effects among females, compared to males. Genotypes of PCSK6 also showed significant sex-limited associations with magical ideation, a component of positive schizotypal cognition measured using the Schizotypal Personality Questionnaire, and total autism score, measured using the Autism Spectrum Quotient. These results partially replicate previous studies on effects of PCSK6 rs1053972 genetic variation on handedness phenotypes, link the PCSK6 gene with the dimensional expression of neurodevelopmental conditions in healthy individuals, and show that associations of this gene with handedness and psychological phenotypes exhibit evidence of sex-limited effects.
Although the mechanisms by which PCSK6 causes lateralization are somewhat understood, the paper says nothing about how it might cause any of the mental illnesses mentioned. So it looks like researchers don't have any clue beyond the observation that [lack of] lateralization is a risk factor for mental illness... except for...
One theory that's been advanced (prior to any genomic knowledge):
T. J. Crow suggested that the genetic variance associated with the evolution in Homo sapiens of hemispheric dominance for language carries with it the hazard of the symptoms of schizophrenia. Individuals lacking the typical left hemisphere advantage for language, in particular for phonological components, would be at increased risk of the typical symptoms such as auditory hallucinations and delusions.
And a 2009 paper that has Crow among its authors claims to
show for the first time how the deficit of lateralization in the schizophrenic brain is specific for the phonological component of language. This loss of hemispheric dominance would explain typical symptoms, e.g. when an individual's own thoughts are perceived as an external intruding voice. The change can be interpreted as a consequence of “hemispheric indecision”, a failure to segregate phonological engrams in one hemisphere.
This study wasn't cited much, so I'm not sure how conclusive its evidence was considered by others.
A year later, a paper (in a more prestigous journal) concluded:
As language lateralization was not significantly reduced in the non-psychotic individuals with auditory verbal hallucinations, a direct relationship between auditory verbal hallucinations and decreased language lateralization can not be established at present.