I have the exact same problem as the OP with the account of one of my family members. The problem seems to be the connected account for receiving/sending email.
This family member uses the new outlook.com webinterface which is the same version as the OP has (screenshots from the OP in this post confirm this, the top bar in the webinterface above is black while it is blue in the old version). One time, when she logged
in, she was asked if she would like to switch to the new outlook.com version and she agreed. From this moment on emails we're not stored in the sent items any more.
Other family members including myself use the old(er) outlook.com webinterface (blue top bar), also with a connected account. We do not have this issue. All family members use the same external SMTP server with the same settings (user/pass, port, etc.) so
I am confident the issue is due to the new outlook.com.
Can Microsoft look at this issue? Is it possible to fix it or perhaps migrate users back to the old version untill this is resolved? Other solutions that are available?