If you are male living together with a woman and have a child, what rights do you as an unmarried father have in Texas? To be succinct, the answer is "NONE", without a court order. Even if the father is designated as such on the birth certificate, the father has no enforceable rights. The child's mother is completely within her legal rights to refuse to allow the father to even see the child, let allow visitation or care. This is also true for the father's entire family, so that any grandparents, uncles, aunts, cousins, etc., would also be unable to have any access without the mother's express permission. However, even if that permission were to be granted, it can be withdrawn arbitrarily at a moment's notice.
While living together is growing in all age groups, young people are clearly the leaders. And we have lots of young people around Austin. According to the 2010 census, the 18-34 age group constituted 22% of the U.S. population, and 25% of Texas population, it is a whopping 35% of Austin's population. The majority of young adults in their 20s will live with a romantic partner at least once, and more than half of all marriages will be preceded by cohabitation. I use Austin demographic data as it is a little easier to come by but you can imagine Round Rock and the surrounding communities are similar.