Hello Emma Yoyo
1.It is recommended to run Set-AuthConfig -PublishCertificate and Set-AuthConfig -ClearPreviousCertificate
2.Personally it is not necessary to restart.
3.To delete the old certificate, you can do so in the Exchange Admin Center (EAC) or by using the Remove-ExchangeCertificate cmdlet in the Exchange Management Shell.
Kind Regards
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