I don't think it does carry over into the broader context of community, generally speaking. I've got a Q&A addressing some of these issues:
Women in Secular Leadership. The authority structure between Adam and Eve was primarily familial. Insofar as Scripture speaks of women outside of Israel's tribal society, in which the social and familial environments are largely one and the same, women are not obligated to submit to men other than their fathers and husbands.
But if by "community" you mean "the church community," then the argument is somewhat different. I won't pretend that I could have predicted Paul's argument in 1 Timothy 2-3, but it does seem to me that Paul saw something in the creation story that made him reject women's ordination. That means that there is something in that story that should cause us to come to the same conclusion.
In any case, it is worth distinguishing between Adam's authority over Eve and the authority of elders over the church. Significantly, elders rule over men and women in the church, so that their authority is not gender-based. So, it would seem to me that the authority of elders is not based on the fact that they are to male. Besides, males don't have inherent authority in the church. They only have inherent authority in marriage.
So, the authority of elders must be based on the fact that they are Christ's delegates. Being male is one of the normal qualifications for appointment as Christ's delegate, but not because being male entails inherent authority that is employed in the office of elder. Elders do not give authority to their office; they derive authority from their office.
According to Paul's argument, as I understand it, the point of ordaining men is largely one of propriety in the eyes of God. In other words, since households are generally to be run by men (recall my earlier response), it is most fitting that the household of God (the church) also be run by men. If there aren't any qualified men, the church still needs leaders, so we should ordain women.
And given the vast weighting of leadership positions in Scripture in favor of men, we should not conclude that there were no women who were more qualified than the men. Rather, the Scriptural pattern seems to be that once you run out of qualified men you start looking for qualified women. This means that the first woman to get the job should be far more qualified (morally, intellectually, etc.) than the last man to get the job. Still, for reasons of propriety in God's eyes (not society's eyes), the job should be given to a man if possible (cf. Num. 27:1ff.).
But note that Adam being created first is only half of Paul's argument; the other half is that Eve was deceived. Both of these details work together to support his conclusion. Frankly, I'm not persuaded that I've got all the kinks out of my doctrine at this point, but it seems to me that Paul appeals to Eve's deception and sin not to say that women usually make poorer leaders than men, but to say that her failure was somehow iconic, so that to maintain proper order after her sin it is necessary to prefer male leadership. In other words, male leadership in the household of God is not based on the present gifts and abilities of the church, but on the fact that Adam was a better leader than Eve.
Now, many will debate that last point, saying that Adam was not a better leader than Eve. But looking at Paul's evaluation and argument in 1 Timothy, it appears to me that Paul believed Adam was a better leader than Eve. It also appears to me that his evaluation formed the basis for his teaching that the church should prefer male elders.