MT and EE allow me to setup paths with zero manual labor. It's the way an application is designed.
No, they don't. At most, MT and EE can allow you to adjust paths inside their own directory. But if I have MT setup at http://example.com/mt/directory, then it absolutely cannot pretend to be http://example.com/vault without me doing some manual intervention to make it work.
On a side note, it's entirely possible to make WP do the same thing here. A minor bit of screwing around with the Permalinks settings can create "pretend" directory paths underneath the WordPress root directory. But only underneath the existing directory, you can't move up in the path without actually *being* up further in the path.
In other words, if you had installed WP at the root directory to begin with, then making it pretend to be in /vault would be just a minor matter, entirely doable without moving files around. But since you installed it in /wp to begin with, everything must have /wp in front of it. See?
If copying a file to a location and modifying a line in it is required, then it surely does not need manual intervention. Many WP plugins themselves do this all the time.
Of course it does. Why in the world would you want the main web directory to be writable by the web application itself? That's a giant gaping security problem if I've ever seen one.
Okay, so yes, WordPress does write to the .htaccess file when Permalinks are changed. However, it always writes the same thing, and really is expecting to not be able to do it, because it will give you the text to put into .htaccess manually as well.
Anyway, yes the file exists. I can see the wp-admin now but the http://domain.com/vault still does not work. I have changed the permalinks to reflect the vault etc. What else should I check for errors? There is no "error_log" file being generated.
Does the /vault directory exist? Did you put the WordPress root directory files into there as explained on the codex page given above?