Yes they do. Just grep the paper for the word "control" and you can quickly find the answers to these questions. This isn't a new finding and they've controlled for all the obvious factors.
I did grep through the paper, and ... no not really.
"Urban" and "rural" never appear. Among uses of "control", for their own work they have controlled for sex and race. Their Table 2 and Figure 2 are the same data displayed different ways and from the same analysis. In their mentions of other work they note related work on Europe that controlled for religiosity, education and income, and socioeconomic and attitudinal factors.
Their table 1 actually shows that the p-value for religious attendance, education, two race variables , and birth year, were all tied with the politics variable, and showed that education, and both race variables had larger-magnitude estimates and z-values.
After reading some of the paper, while I still think there's a relationship between political beliefs and number of kids, I don't see that their methods are convincing that this is more important than other such correlations.
"Urban" and "rural" never appear. Among uses of "control", for their own work they have controlled for sex and race. Their Table 2 and Figure 2 are the same data displayed different ways and from the same analysis. In their mentions of other work they note related work on Europe that controlled for religiosity, education and income, and socioeconomic and attitudinal factors.
Their table 1 actually shows that the p-value for religious attendance, education, two race variables , and birth year, were all tied with the politics variable, and showed that education, and both race variables had larger-magnitude estimates and z-values.
After reading some of the paper, while I still think there's a relationship between political beliefs and number of kids, I don't see that their methods are convincing that this is more important than other such correlations.