Hello @Sergey,
According to the error message you provided, this Error "Load balancing failed to find a valid mailbox database" usually occurs when enabling or creating a mailbox. The possible solution is to modify the parameters. First, as you posted, you need to run the Get-MailboxDatabase -Identity "<mailbox database>" | Format-List Name,IsExcludedFrom*,IsSuspended* command to determine whether the database is locked. If the result shows True, you could change the setting to False by running Set-MailboxDatabase "<mailbox database>" -<parameter> $false. Also, remember to always keep your Exchange Server up to date and make sure you have any Exchange Server cumulative updates installed.
About owa is also not open and Microsoft.Exchange.Data.Storage.UserHasNoMailboxException occurs, this error may occur if the user's mailbox setup is not completed, and if the license has just been assigned to this user, it may take more time to mailbox set up. Alternatively, if the problem persists, you could try disabling and re-enabling OWA for your mailbox, clearing cache, reopening the browser and logging in again.
Hope the above information is helpful to you!
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment". Note: Please follow the steps in [our documentation] to enable e-mail notifications if you want to receive the related email notification for this thread.