Jewish Law

If your wife  permanently lacks the legal capacity to consent to divorce that is required under Jewish law to do so, you will not be able to divorce her, but you can apply to the rabbinical court  for permisison to take another wife.

Not if  you prove she was having adultery, if she does not consent to divorce, or , if you make a divorce agreement in which she agrees to forego her Ketuba, and you both avoid the issue of her adultery, as mutual consent is sufficient grounds for divorce. Under Jewish law a fornicating wife is known as a "Moredet" ('rebellious wife') , and loses her right to her Ketuba, (sum stated on the marriage bond/contract) and maintenance during the marriage. Only the rabbinical court can declare a wife suspected of adultery a 'Moredet' .

Under Jewish law, as a divorcee you cannot marry a man who is a 'Cohen' i.e. whose surname is Cohen, and who is a descendant of the ancient priestly tribe of Cohens. Apart from that, you are free to marry another Jewish man.

There is a way round the bar on marrying a Cohen under Jewish law - you can cohabit , or marry in a civil ceremony.

No - because they are mistaken about the law.  A Jewish couple who marry in a civil ceremony will not be risking stigmatizing their children by not marrying according to Jewish law. Their children will have normal status under Jewish law, as it is assumed that they are 'dedicated' to one another and that there is 'Kiddushim' (dedication).

No, contrary to popular myth! Where a Jewish couple live together and have a child then there is the assumption that they are 'dedicated' to one another, even if they are not officially married, and the child has normal personal status.