I’m not sure I can help with all these issues since they might not be linked to Jetpack itself, but I may be able to help you understand the false site down messages.
Jetpack Monitor checks your site every 5 minutes and try to load it. If it can’t load the home page is less than 10 seconds, it considers it as down and sends you an email.
So if your site is up, but too slow, Jetpack will consider it as down anyway, and will trigger a false site down message.
You can read more about it here:
https://jetpack.me/support/monitor/#site-down
Site goes down frequently,
[…]
site defaults back to former site, etc.
When your site goes down, do you see an error message instead of the site, a blank page, or anything else? How do you usually bring your site back up?
When it defaults back to the former site, is that a site that’s hosted somewhere else, or a different theme on your current site?
Could you also post your site URL here, so I can have a look?
If you want it to remain private, you can also contact us via this contact form:
https://jetpack.me/contact-support/
