Apple Mail filtering on its own

[tape]

I recently changed my email address associated with a Costco/medical site to a .ME account from the original POP. I stopped getting responses from them. After calling and calling, I finally got to a tech person on their side and they said that apple was probably filtering the mail thinking it was junk. But the messages never showed in the junk folder. I checked to see if the Costco/medical site's email address was in my contacts--it was. I changed Costco email association back to my POP account and the emails started coming through.
The other day, I received a number of emails to a gmail account I have only on my mobile devices. I wanted to see the emails on my iMac, so I dragged them to my .ME account which is also on mobile devices. They never showed in the .ME account. I backed up and relocated the messages on my iPad and then forwarded the messages to my .Me account from the original gmail account. Nothing showed. I then forwarded them again to a .LIVE account and they showed.
I checked the preferences in apple mail and find no rules that would disallow me to receive emails to my own account from my own address. How do I get apple mail to deliver all my mail?

 

[MacInWin]

What is a ".ME" account? Do you mean "me.com?" The Costco/medical site tech guy was blowing smoke. Apple doesn't "filter" anything. And what is a ".LIVE" account? Your post was very confusing, sorry.

 

[tape]

.ME is me.com from apple and .LIVE is live.com from microsoft.
Another incident today: difficulty logging on to Creative Cloud from Adobe and was forced to change password. Primary Adobe account has my apple email and so a request was sent there, except it never arrived. I signed in again, but with my secondary email, which is my live.com. That showed in my inbox almost instantly. Getting concerned with apple mail since it is my main email address

 

[MacInWin]

I would check all of the me.com account settings in Mail. It would appear that you have something not set up properly. Use Mail Preferences for the details on the accounts.