Partha worshipped the brahmanas and went to Droupadi. Then, out of love, Droupadi told Kuru’s descendant, “O Kunti’s son! Go to the daughter of the Satvatas. A second load always loosens the first tie, however strong.” Krishna thus lamented in many ways. Dhananjaya pacified her a lot and asked for forgiveness. He quickly went to Subhadra, who was dressed in red silk. Partha sent her away to dress in garments worn by a cowherd lady. But the illustrious one looked beautiful even in that attire. Arriving in the best of houses, the famous and beautiful Bhadra, wife of a warrior, with copper-red eyes, paid homage to Pritha. Then, with a face as radiant as the full moon, Bhadra quickly went and paid homage to Droupadi, saying, “I am your maid.” Then Krishna arose and embraced Madhava’s sister and lovingly said, “Let your husband not have a rival.” With a happy heart, Bhadra replied, “May it be that way.” O Janamejaya! The maharatha Pandavas were then happy in their hearts. Kunti was also extremely delighted.
Source: Bibek Debroy's English translation of the Mahabharata Critical Edition (BORI).
That is the entire passage as is. It is clear that Draupadi was upset. She had her heart set on Arjuna but was let down. She realized that it wasn't like Arjuna could un-do his marriage with Subhadra (nor did he have any desire to do). It was what it was and she had to live with it. Secondly, she was married to five men, so marriage was a nightmare for her. Furthermore, all of her husbands took up wives, and therefore, she had numerous co-wives. It was an everyday reality for her.
It would seem that, during the time of antiquity, life was quite painful for women everywhere.
In so far as your final question is concerned: the first wife, also known as the head wife was the Head Wife. During the Ancient times, the status of Queen was given to the Head Wife (usually the first and eldest amongst her co-wives) in order to clarify lines of succession. Could she present orders to her King? Absolutely not. Did she have a limited degree of power to exact over her co-wives? Absolutely. Keep in mind that a co-wife could always register a complaint with the King, which is what Kaikeyi did.
I hope that my post shined some light on the topic. Longtime lurker here :)