In Ohio, these matters are all part of the jurisdiction of the Probate Court, a division of the Court of Common Pleas. In Franklin County, the local Probate Court can be reached through their portion of the Franklin County Governmental Portal Website, located here: Franklin County Probate Court. The phone number for the Probate Court in Franklin County is (614) 525-3894. The Probate Court is located on the 22nd floor of 373 S. High Street, next to the Municipal Court Building. Their site also has the information about the online application for a Marriage License.
Mostly, these matters are the jurisdiction of the Division of Domestic Relations of the Franklin County Court of Common Pleas. Marriage Licenses are part of the Probate Court. Inquiries regarding these matters should be directed to the appropriate divisions of the Court of Common Pleas. Child support issues, likewise, have an agency that handles these issues. The Child Support Enforcement Agency (CSEA) falls under the jurisdiction of the Franklin County Commissioners' Office. Below are links to these three agencies.
The other area concerning these matters that we receive questions regarding is about the availability of personnel to perform wedding ceremonies at the Courthouse. The performance of marriage ceremonies is delineated in Local Court Rule 2.01 and the process is outlined here on the website under the General Information Page under the Services section. Information regarding scheduling a wedding ceremony can be obtained by contacting Court Administration at 614-645-8214, but it is a service that is dependent upon the availability of personnel, so scheduling in advance is of primary importance and not everyone's request can be accommodated.
While the traditional concept is that you go down to the courthouse to get a copy of your birth certificate, here in Franklin County these matters are handled by the Columbus Health Department. Here is a link to the page that they have explaining the process they have in place for these requests, Getting a birth or death certificate.