I can not explain ANY of the issues you noted given the details so far. But I would head to gmail.com on a web browser and try a gmail account next without using ANY MAIL APP!
*Mails have not been received: For weeks on end during the last few years. Then suddenly, they arrive again, but not the ones I didn't receive. Sending has worked ok.
*Mails disappear: When I open them on my iPhone, being close to my iMac, they sometimes suddenly disappear. In fact, all new ones vanish, both on the Mac and on the iPhone. I suspect interference from radio waves.
*Anyone sending me mails gets an "undeliverable" message, but the mails arrive!
Is there anyone out there who can explain these problems, or suggest how to solve them?