For the specific question, FIMSynchronizationService working or not does not affect performance of SharePoint. It doesn't, however, mean that same issue causing this specific error wouldn't cause slowness, but it doesn't seem likely (considering the following chapters below).
FIMSynchronizationService is responsible for synchronizing user profile properties from Active Directory to SharePoint User Profile service and runs in the background.
What has happened is that someone has most probably Started or Stopped the FIM Windows Service manually using Services application in Windows. This should not be done as SharePoint wants to control this and only starts/stops the two services when the Synchronization is configured successfully and UPS service is started/stopped from SharePoint Central Admin.