How to place custom menu in user profile
I have been searching for this for 2 days now, and decided to go for bounty. But, didn't know the solution was so simple. I tried doing the same on fresh install of Drupal and it worked fine. And kept on adding the modules one by one, again testing the functionality for any change.
It appeared to me that after disabling Menu token module, the requirement worked absolutely fine.
Then I checked to see if I gave a wrong token replacement. And the menu structure which i was using was as below  

--My Account with path [node:url:absolute] which i thought would direct to same page.
----Edit Account with path user/me/edit
----Edit Account with path user/me/edit-profile
----Logout with path user/logout  

And I used gave the path as *user/** in the Visibility settings of the block(highligted of corporate theme) as discussed in question and comments above,

The Problem identified(as per me) was that I was considering user profile page as a node which I think now is wrong, so the token [node:url:absolute] was actually a wrong token and hence the menu was not displaying on user page. What I did is changed it simply to [current-page:url] (as I wanted the user to be on same page, and I suppose we cannot give # in the path of the menu to achieve it)as shown in fig below and used the same visibility settings and it worked :)
So, basically the parent menu path was wrong which created the problem. Nevertheless, it was good learning experience!!