Is there a specific supplication to be recited when performing ghusl after menstruation or childbirth, or is the ghusl itself sufficient, and does it entail two rak'ahs of gratitude for purification thereafter?

There is no specific supplication designated for ritual bathing (ghusl), except for the desirability of saying "Bismillah (In the name of Allah)" at the beginning of the bath. As for performing two rak'ahs of prayer after ghusl, there is no harm in it, based on the hadith of Bilal in the Sahihayn, where he said: "I have not done any deed more hopeful for me than that whenever I purified myself (performed ablution or ghusl) at any time during night or day, I would pray with that purification what Allah ordained for me to pray." And purification (tahur) includes both wudu (minor ablution) and ghusl (major ablution).