When I run the Domain Check in Data Review, it does not give me the option to Write to Reviewer Table, only to browse the results . How do I correct these Domain issues?
Sounds like you may need to start a Reviewer Session in order to persist your results in a geodatabase. Here is a link to the help topic which outlines how to start a session: http://desktop.arcgis.com/en/arcmap/latest/extensions/data-reviewer/starting-a-reviewer-session-in-a-nonenterprise-geodatabase.htm
Once you have a session started, you will have the option to either interactively browse the results or write results to the Reviewer Table after running a check.
I have started a session and when I try to run a check, such as the Domain Check, it will not populate any of my feature classes within my Geodatabase. I have also contacted ESRI Tech Support and never received an answer.
Are you running the check from a Reviewer Batch Job or from the toolbar? If from a Reviewer Batch Job, then it should automatically write the results to the Reviewer workspace. You can access the Results from the Reviewer Table. If you are running the check from the Reviewer toolbar, then you will be prompted when the check is done if you want to review the results in the Browse Features dialog or Write them to the Reviewer Workspace (as Jay had mentioned). Make sure you select the latter option.
Retrieving data ...