Must a husband divorce his wife Islamically after she has been granted a judicial divorce from a non-Islamic court at her request? Is he entitled to seek compensation for the khulʿ? And is she entitled to a share of his property?

As long as you have not uttered the words of divorce to your wife, nor written a divorce with the intention of effecting it, your wife remains under your marital bond. Divorce does not occur merely by its issuance from man-made courts. If you do not wish to return to her, it is preferable that you divorce her, for separating from a woman of bad character is commendable. You may also refrain from divorcing her until she redeems herself from you with money. She is not entitled to share your wealth without just cause, and there is no consideration for any law that permits her to do so, as it contradicts Islamic law.

You both can refer the matter to a trustworthy body among the authorities specialized in Muslim affairs, as it acts in place of an Islamic judiciary.