Run in Safe mode does several things.
One very important thing is does is Eliminate Third Party Applications, extensions and drivers from loading.
Judging from your observations above, running in Safe Mode may indicate some piece of Third Party Software is interfering with either the Internet Connection and / or the normal functioning of Mail .
Download the Application Etrecheck directly from the Developer.
The application is free or paid from added features.
It will take a Snap Shot - both the hardware and software.
The Report will Not Reveal Any Personal Information.
Post back the Full Report - copy and paste - >>>> using the Additional Text Icon ( 3rd Icon to last ) <<<<
We can have a look at the report for possible issues and may have possible suggestions to resolve the issues.