Hi,
I'm understanding that alerts/reminders exactly like in Outlook.
In browser with js script, you need to refresh the page to make the code running. The only choice here is make an function and execute it every interval time.
Form level: you can use support method Xrm.Page.ui.setFormNotification("....") --> It's OK.
Site level: I never see anything SDK that we can put custom jscript in site level. So dunno if it works.
Just some of my thought, hope it helps.