Is there a method to log the errors that Salesforce marketing cloud displays while doing a Subscribe and Preview in a Test send?
Unfortunately, there is no way to store the logs when you preview an email. Yet, what you can do is:
1- When an email is failed while sending, you can ask Salesforce Support to get the logs for this particular email. You can also do that on your end - again after you send the emails as a test or actual sendout - with the use of RaiseError but you need to specify your errors: https://developer.salesforce.com/docs/atlas.en-us.noversion.mc-programmatic-content.meta/mc-programmatic-content/raiseerror.htm
2- You can check and log the problems manually by creating a Cloudpage and following this way: Error Handing in Ampscript