Cannot get Gmail emails to populate iCloud

Question

I have tried every combination that I can think of, but I cannot get my Gmail messages to be received by my iCloud account.
All of my messages on my Mac are in my Gmail account properly and I have no problem sending and receiving these.
I set up the "iCloud IMAP" account with the following setup:
Account Information:
Checked Enable this account
Description: iCloud
Alias: Custom
Email address:  [email protected]
Full Name:  HH
Incoming Mail Server: p04-imap.mail.me.com
User Name: xxxxxxxx
Ougoing Mail Server (STMP): iCloud (iCloud)
Checked: Use only this server
TLS Certificate: None
Advanced:
All items checked
IMAP Path Prefix:  (blank)
Port: 993      checked: Use SSL
Authentication: Password
checked: Use IDLE command....
================================================
Using the drop down menu of the Outgoing Mail Server, the following information is:
Description: Gmail       Server Name:  smtp.gmail.com       In use By Account: HH-Gmail
Account Information:
Description:   Gmail
Server Name: smtp.gmail.com
TLS Certificate: None
Advanced:
Checked Use custom port: 587
checked: Use Secure Sockets Layer (SSL)
Authentication: Password
User Name: xxxxxxxxx
Password: ...........
==========================================
NOTE: EVERY ITEM SHOW ABOVE IS ACTIVE. THE ONLY ITEMS THAT ARE GRAYED OUT ARE THE "INCOMING MAIL SERVER" AND THE "USER NAME" IN THIS SECTION. I DON'T KNOW ANYTHING ABOUT "p04-imap.mail.me.com", BUT I SUSPECT THAT THIS IS THE CAUSE OF NOT BEING ABLE TO RECEIVE ANY OF MY EMAILS IN iCLOUD. THE SAME CONDITION OCCURS IN MY iPHONE AND iPAD WHERE IS SAYS IN iCLOUD "NO MAIL".I HAVE TRIED CHANGING THE PORT NUMBERS, THE PASSWORD, THE ALIAS, BUT NOTHING SEEMS TO WORK TO BRING IN THE EMAILS.    ANY SUGGESTIONS? THANKS.

Answer 1

lCsound1,
Thanks for the reply. This is a real surprise. Can you elaborate a little more on thisi issue.
I am not sure I fully understand iCloud to start with.
When I open my iCloud from a web browser, I do see, assuming some of the items that are checked in the Internet Accounts window such as Contacts, Calendars, Notes, etc. But, when I open the Mail, nothing which is what you mentioned in your reply.
When I open the Mail/Window/Connection Doctor, it shows the status of my Gmail (SMTP), my HH-Gmail (Google IMA), and iCloud (iCloud IMAP) are green and "Connection and longin to server successful"? What does this mean?
In general, does the "Incoming Mail Server: p04-imap.mail.me.com", being grayed out mean that nothing can be brought into the iCloud account?
While I have the Documents and Data checked in the Internet Accounts window, it does not show up as an icon in my web browser, iCloud.com account. How is this handled by iCloud?
Thanks a lot for your help in this matter. I have never seen your reply comment in any of the Forum threads nor on any of the Apple docs that I have been able to pull up.