Is it permissible to supplicate with specific supplications for facilitating marriage—such as abundant seeking of forgiveness (istighfar), reciting Surahs Al-Zalzalah, Al-Kafirun, Al-Nasr, and Al-Ikhlas, performing two rak'ahs of prayer followed by supplication, and imploring God with verses and supplications like: "My Lord, indeed I am, for whatever good You would send down to me, in need" and "O Allah, grant me from Your presence a spouse," and other similar things mentioned in the question—or is it sufficient to make a general supplication for a righteous spouse?

It is permissible for a person to supplicate to Allah with whatever he wishes of the good things of this world and the Hereafter. Supplication is generally encouraged, whether it is a prescribed supplication or not. There is no harm in supplicating with the mentioned supplications, as they fall under the general command to supplicate. However, it is better to avoid specifying things for which there is no textual evidence from the Sunnah (definition), such as: frequently reciting certain surahs with a specific intention, or supplicating with "O Allah, by the right of Your saying..." and mentioning specific verses to ask for a specific matter. As for frequent seeking of forgiveness (istighfar), it is beneficial for attracting provisions and relieving worries.