An easy way to determine if your Fastmail account is working properly is to find another computer connected to the Internet--possibly at work, or from a friend, or at the library. Open up a mainstream browser like Chrome or Firefox and go to the website
https://www.fastmail.com/. Log in using the button in the upper right corner of the web page. Do your missing emails appear there? If so, the problem is with how your iPad or iMac email programs are set up.