Is the aforementioned 'urfi' marriage valid and legitimate in the sight of Allah, and is it forbidden to not have children due to special circumstances?

The majority of scholars are of the opinion that marriage without a guardian (wali) is invalid, and this is the relied-upon view. Accordingly, this woman must be divorced immediately due to the invalidity of the marriage. If you wish to marry her validly, it must be through her guardian. If the guardian refuses to marry her to you while you are a suitable match, she has the right to refer the matter to the Sharia judge.

However, if you contracted the marriage with the woman believing in its validity, following the Madhhab of Abu Hanifa, then your marriage is valid.