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.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!