Regarding error ""System.DirectoryServices.DirectoryServicesCOMException (0x80072030): There is no such object on the server."  This might be related to the issues discuss on this other thread, the post include different scenarios as well as suggestions.  In case the posted suggestions did not address the issue please submit a case with Technical support and submit a copy of the Directory Server event logs (unfiltered) as .evtx files:
Applications and Services Logs > Laserfiche > Directory Service > Server > Operational trace
Applications and Services Logs > Laserfiche > Directory Service > WebAdmin > Admin
Applications and Services Logs > Laserfiche > Directory Service > WebAdmin > Operations

In regards to the second error
System.Data.SqlClient.SqlException (0x80131904): A network-related or instance-specific error occurred while establishing a connection to SQL Server.

the issue might be related to a corrupted License Site and/or corrupted License database, Please open a case with Technical support and provide a copy of the Directory Server event logs. Also confirm if there are any issues updating the master license database , refer to kb 1013686

https://answers.laserfiche.com/questions/88453/LF-Server-10-cannot-connect-with-LFDS?sort=oldest#replies