501c3 status is in regards to campaign contributions so would not be effected.
"Currently, the law prohibits political campaign activity by charities and churches by defining a 501(c)(3) organization as one "which does not participate in, or intervene in (including the publishing or distributing of statements), any political campaign on behalf of (or in opposition to) any candidate for public office." As long as the church doesn't fund a particular candidate or political party with church money, or support a party or candidate with direct statements from the pulpit, in publishing, etc.. they don't lose their status. For example: The Catholic church in your area publishing in the local paper: St. Luke's urges all of it's members and any catholic to vote for jack Daws for dog catcher because he's a good, upstanding catholic." That would be cause for investigation.
Like Shjade said.
Allowing someone to participate in a wedding ceremony is up to each religion. One can't demand to be married in a church because they wish it to be so. Most religions have requirements that must be met before two people can be married in their religion, a ceremony that is specific in the eyes of that religion but has no legal bearing.
It's the paper's registered with the court house that count.