Marriage between first cousins was not uncommon in the United States and Britain then, and indeed until quite recently (probably in the last hundred years.) Even in the nineteenth century, though, thirteen would have been considered very young to be married but it was by no means unheard of. I can't remember exactly what the youngest age for marriage was for girls during that time, but traditionally the Catholic church in Europe had designated the age limit at 12 for girls and 14 for boys. I haven't done a lot of research into this particular top since my undergrad years, but I'll try to find some sources to post for you if I get time.

It's difficult to come up with specific numbers, as the 1890s are the earliest decade for which we have any concrete evidence of age at first marriage. However, I ran across something which may be useful or at least interesting to you - although it doesn't deal with the cousins issue, it does deal with the "age of eligibility"/marriageable age of young women. Specifically, young women in the 1840s; Poe married his wife in 1835, but seems close enough to give a good comparison. This is an examination of the marital habits of the early Mormons: polygamy with often very young girls. Scroll down to the second article on the page, titled "Was it normal to marry 14 year old girls in Joseph Smith's time?" It discusses age of menarche and what that means for "age of eligibility" in the 1840s; average age of menarche in 1840 was apparently 15 or 16, meaning that "age of eligibility," i.e. age at which childbearing could reasonably begin given the development of the girl's body, could not generally be reached at age 14, so it would be unusual for a 13 year old girl to be considered eligible for marriage. I'm not sure what the legal age of marriage would be, I can't find any sources. However, some states to this day have "legal marriageable age with parental consent" ages as young as 12, so it's entirely likely that the legal age of marriage with parental consent was very young.