I use the Inbox a lot and think that it is such a major feature that it should have its own place in the top right of the screen (between [username] and chat?) This would have the advantage of being able to access the Inbox with one click.
No! The inbox is an often-used feature, so it should have an easy-to-reach position, i.e. a corner. (See Fitts's law.) The middle of the screen is for less useful things.
The top left corner is easier to access than the others, for several reasons, not least that not everyone has a big enough window to fit SE's very wide layout. People with smaller screens will have the right-hand part of the screen obscured.