(Why) Do US studies of violence and mental illness tend to report no link but European studies seem to find one?
I have been away from here for a while as I have been concentrating on my University studies, however, I have just come back and boy have I come back at an opportune moment.  Your question showed a lot of research whilst being put together and this is an interesting topic of discussion as I have been discussing the subject of mental illness and violence myself with a couple of friends.

As you pointed out, Heather Stuart stated that


  Definitive statements are difficult to make and it is equally possible to find recent literature supporting the conclusions that the mentally ill are no more violent, they are as violent, or they are more violent than their nonmentally ill counterparts.


From what I have seen, I would agree with you that this still stands today as a response to your question.  The thing is, I have not seen an article which gives a definitive answer.

It is interesting to read the 2017 study on the FBC, RADAR and ALSPAC datasets you provided.  The thing to note with this study is that although


  In the additional sensitivity analyses, we also adjusted for other factors including parental criminality, conduct disorder, and maltreatment where data were available.


what about information regarding maltreatment etc. which was not available?  Those children who were subjected to undocumented external factors, which could affect their behaviours, would have been excluded from the adjustments and therefore, the statistics presented may be inaccurate.

Another point to make is that as the datasets seem to be based on self disclosures of violence, this, along with other issues raises all sorts of questions.

I would ask how the relevant studies defined violence when asking for the disclosures.  How were the questions put forward?  What was included in their definition of violence?  Was verbal and/or mental abuse considered to be violence or was the scope restricted to just physical violence?

This could be one reason why there is a discrepancy between US studies and European studies.  Another possibility is, were there cultural differences in play?

The differences in all the studies you pointed out using the Elbogen &amp; Johnson data highlights the complexity of such a study.  What mental illnesses were covered in the studies?  Were any of the mental illnesses a result of (or made worse by) substance abuse, and if so, what is the prevalence of this factor?

There could also be all sorts of problems with interpretation of the data.  @ArnonWeinberg shows an example of possible misinterpretation of data when referencing Sweden as the rape capital of the world.  (See here for what I am — and possibly also what Arnon is — referring to).

There are so many questions which seem hard to answer with what I have seen and these sort of questions may not be able to be answered in a full meta analysis involving all studies on the mentally ill as this may end in a great deal of speculation.  However, a full meta analysis of all studies in all age groups could point these flaws out and lead to a more thorough study which would iron out those wrinkles.  I personally would look forward to seeing one.