Thunderbird Solution
FWIW - Based on a suggestion above, I downloaded Thunderbird, installed it and asked it to install the e-mail accounts I have with my cox.net provider - all the problems noted in Mac Mail vanished in using Thunderbird. So - I will probably stay with Thunderbird at least until MAC gets this fixed.
I removed all email accounts (6-10 times), reinstalled them, stopped Mail, restarted my iMAC, did a 'rebuild' and the same problems return. Cox says it is a Mac Mail 'connectivity issue' and Mac says it is an 'Idle timing ......something' at Cox.
Technicians at both Cox and Mac Mail state that the issue is with the other service. Funny about that when Thunderbird works great and the mail appears in both Cox and Thunderbird, just not in Mac mail. Go figure!!!
Bob