Find My Phone and Deleting iPhone
Whatever message you wrote when you put it in Lost Mode will still be displayed after you erase the device, so no they won't necessarily know the difference.
If the phone hasn't been put in Lost Mode and wasn't protected by a passcode (ie. they've been able to keep using it) then yes, of course they will notice if the phone resets and everything disappears.

Whatever actions you take will come into effect whenever the phone gets back online. Your question is a bit unclear so if by "notify" you mean "gives them the option to stop the erase" then no, of course not.

For more see this Apple page.