Is the marital relationship still valid between the questioner and his wife, especially after she requested a divorce, he refused it, and the judge set a period for deliberation?

If you did not pronounce a divorce, your wife is still under your marital bond. Her request for divorce or her recourse to a judge does not make it effective. The husband must provide a dwelling for his wife, and this right is not forfeited by her owning a dwelling. If the wife bought a house with an interest-based loan, she is sinful, but the house is her property, and the husband may rent it from her. As for not having children, this should not be a cause for problems, and the couple should reconcile their differences. It is not permissible to seek judgment from man-made courts; the alternative is to seek judgment from the authorities responsible for Muslim affairs.