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Welcome to the magic of modern browsers. Every so often the browser is trying to refresh the thumbnails cache for the sites that you had recently opened (but closed) or the tabs you still had open when you put the device to sleep. This will happen in the background when your computer wakes up – typically without turning on the screen – to fetch updates (e.g. Connected Standby on Windows or Power Nap on macOS). So, the device you left back home thinking it's in sleep mode? That's what is causing the access.

The solution is the following before you put a device to sleep mode. Close all tabs that had your site's admin open. Turn off the thumbnails for recently visited sites.

I can tell you for a fact that Admin Tools uses the Joomla Mailer API to send an email when it detects something. That's it.

From that point onwards there's Occam's Razor to guide us.

Could your mail server be delaying sending the emails for a week? Sure, it could, but it's extremely unlikely. All emails would be delayed, not just some specific ones. When you start needing to devise an incredibly complex explanation to fit your observations the chances are your explanation is wrong (Occam's Razor).

Regarding your idea that a hacker might do that... This doesn't hold water. They'd have to come in from your home IP when you say that your device is powered off. So what are we suggesting here? That a hacker magically knows when you're not at home, breaks into it, connects to your router, logs into your site, does nothing to your site at all and logs out just to mess with you? That sounds a bit strained, doesn't it?

If you are unsure, here's a simple test. Before leaving your house disconnect all network cables and turn off the WiFi – or, better yet, turn off your Internet router. These problems will disappear, proving the point.

Maybe a miscommunication. I said I can't imagine a hacked would do that.

All right. It sounded like you said the exact opposite. I understand now.

And most likely they have send that email to me after a week. 

That is a far more likely explanation.

The Admin Tools email doesn't show the time of the login. Only the time when the email is send.

They are the same. The email is sent when the login takes place. The time that the login occurs is the time the email is sent through the Joomla API. The actual email leaves the mail server a few moments to a few seconds later, i.e. the time it takes for Joomla to connect to the mail server. During that time Joomla "hangs" waiting for the mail server reply. So I'm pretty sure it's not a week ;)