I've never had Wordpress email me on a 404 error, so I'll guess that it's a plugin acting up -- or maybe even the theme, if you didn't have the problem before changing themes.
Are the emails coming in fast enough the you could quickly tell if the problem is fixed? If so, I'd disable all plugins and see if it stops. If it does, enable them one at a time until it starts again.
Once you hit the one that kicks the problem in, you could also disable all of them again, then just re-enable the one that caused the problem. If that doesn't bring the problem back, maybe that plugin is clashing with another, so you could leave it on and activate them one at a time again until you see which two aren't playing nicely with each other.
If disabling the plugins doesn't help, I'd switch back to the default theme and see if that does it.
Hopefully that will help narrow it down!
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