When I sent Emails via LO Write I do not see these mails in my sent folder …
Then you seem to talk about Mail Merge functionality, which doesn’t involve your email client at all, since the emails are sent directly via SMTP. Only using File -> Send -> Email Document...
would involve your email client (and would work only, if the email client supports Simple MAPI)
To enhance logging of `mailmerge.py` (which is the Python script finally doing the dispatch) you'd need to modify the script `mailmerge.py` (which resides in the `../program` folder of your LibreOffice installation) and set `dbg = True` (but hard to tell, whether the script gets any notification about invalid addresses from SMTP server - I'd not expect such information)