Why is covering your hair only applicable to married women while all other halachas in tznius apply to everyone?
This is a good question, and in fact, it was the custom for certain communities that even unmarried girls went with hair covered. This is also ruled by the Shulchan Aruch (Even Ha-Ezer 21:2), according to various commentaries (see Beis Shmuel; Magen Avraham 75; Mor U-Ketzia, 75).
However, the almost universal custom is that unmarried girls do not cover their hair, and it appears that this follows the “custom,” which was that married women cover their hair, and unmarried girls don’t. It is hard to know how this custom evolved, but once this became the custom, there is no longer a prohibition on unmarried women to go with uncovered hair, for this is not a “covered place” in the body for unmarried girls.
This general custom is mentioned in the Shulchan Aruch (Orach Chaim 75:2, with regard to reciting Kerias Shema), and it is clearly an ancient practice.
[Rav Ovadyah writes in a teshuvah that even for Sephardim (the custom in many Arab lands was for girls to cover hair even before marriage) the custom is to be lenient. He also mentions that he asked Rav Elyashiv about this matter, who answered that we are lenient concerning unmarried girls.]