Hi Joseph,
Thanks very much for your suggestions. I have spent the entire day on this. Sometimes it seems to be better, other times not. Weird, I know. My Mac has some different mail settings than my iphone and ipad. For example, the port numbers are different. I changed them and things got worse. My mail account went "offline" and I could not send emails. I changed them back and those problems, and the original one I wrote about, improved. I am going to keep playing around and see what happens. I will next remove the Mac mail account and then add it back.
As for your three questions, I am using Gmail. I tried the folder question but it is hard for me to know for sure. My folders are not set up the same on all three devices. Yes, I am using the built-in mail app on all devices.
I tried to set up an exchange account, which has been suggested in other places, but I was unable to do it.
FWIW, everything else syncs perfectly--Pages, music, calendar, etc.
Thanks again. I will get back to you.