DO use a your own domain. Advantages far outweigh the disadvantages. Instead of being at the mercy of whatever email provider you choose, you're free to move your domain where ever you want, without updating "a million" senders. DO NOT host your own email server though. A couple of decades ago it made perfect sense to do so, but these days it's simply not worth the trouble. Instead use Google/Outlook/Apple/Whatever t…
For backup, I have tried quite a few, and the simplest I found (for now) is just to run Thunderbird (daily, weekly) and set it to download a local copy via POP. MBOX is an open and acceptable format by most other clients (if you need to change clients).
I will continue to clean up and streamline the processes, but the general idea is to have a backup (preferably local) somewhere, while the primary email can be accessed from your choice of client or the web.