Email support

No, Mail on a Mac does not have receive receipts you will see in the Messages app.

The grey curved leftwards pointing arrow will be seen in the In box. It is an indicator that you have set a reply to the item visible in the In Box. If you click on that arrow Mail will show you the mail item sent in response.

If you see a mail in the Sent box it is the part of the correspondence which has been sent (by you) and therefore needs no indicator that it has been sent. It is implicit in being in the Sent box.

Occasionally see a rightwards pointing arrow and these are indicators that your incoming mail has been forwarded to a different email address.

The convention is left goes back to the originator, and right goes onwards to a new recipient. The model is taken from the way we read text from left to right. I guess this is changed in languages where text is read from right to left, not to mention vertically! Is it Japanese which is read top to bottom in columns then right to left across a page?

If the message has not been delivered to the recipient, you will receive a message from your email client server.

Have never seen a grey curved arrow on sent e-mail messages here.

That's a sign that you've replied to an email.

Did you click/touch on Reply and is your message in your Sent box? If you created a new message it is not treated as a reply because it is new. Or perhaps it failed to send?