A complicated issue to do with email on my Mac and on my iPhone

I have tried for days to sort out the email problems I'm having using written advice online, but with no success. On Thursday I began a Chat on Apple Support but it seemed an inappropriate medium for dealing with the complexities of the case. I really wanted to explain all the facts and then fill in any gaps for the adviser, but the Chat system doesn't seem to work that way.

I have tried to set out all the facts below and I'd be very grateful if anyone can see a way through it all as I am getting more and more confused.

BTinternet email account

At bt.com I can access my BTinternet emails using password xxx to login to my account. I can send and receive emails there.

On my iPhone I can receive BTinternet emails but not send them. When I try, I get this message:

Cannot send mail

The username or password for “xxx@btinternet.com” is incorrect

OK    Settings

Needless to say, the username and password are the same as those I successfully use at bt.com.

On my Mac, a few days ago I kept getting this message when I tried to send emails and so, following advice available in various places, I deleted the BT internet account in Settings and attempted to add it back. But I then got this message: 

Unable to verify account name or password.

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Google email account

When I try to send a message from my Mac via Gmail I get this

But I can send from the iPhone via Gmail.

I have just tried to send a message from my Mac using my iCloud address and it is still in the Outbox. 

After several minutes this message appears

I have tried clicking Try with selected server and highlighting iCloud and then Try with selected server. But still the message remains unsent.

I have just tried sending from my iPhone, using my iCloud address, to my iCloud address, and the message got through to the inbox on the phone and the Mac.

The messages like this one The SMTP connection to server “mail.btinternet.com” failed … Try with selected server are hard to get rid of. If I click Try later they go away for a few minutes but then come back and I have to click Try later again, but still it only goes away for a short time. 

A few days ago I had a Chat with Apple Support during which I tried to explain my issues. the adviser asked me to set up a test user account on my Mac, which I did.

A few days ago I had a Chat with Apple Support during which I tried to explain my issues. the adviser asked me to set up a test user account on my Mac, which I did.

I was able to add my Gmail account in Internet Accounts, but when I tried to add my BTinternet account I was unable to with (*I think) this message Unable to verify account name or password.

*I took a screenshot and emailed it to myself and then logout out of TEST and back into my own name on my Mac. But when I tried to upload it here the filename was greyed out.