Why does my application behave differently with Unity?

I noticed in Unity that the menu bar seems to be "redirected" to another place outside of my application GUI, to a bar at the top of the screen.

I think what you are refering is the default behavior on Unity. Unity uses a global menu (the one you see at the top) similar to what is seen on OSX.

If you want this can be disabled in your application, but I think that is not very recommended: anyone using Unity will expect the menu not on the window, but at the top.

Regarding the entry box I would guess that it may be a problem with the Ubuntu theme. To check it try to change the Ubuntu theme (under "System Settings" -> "Appearance") and see if the box stays the same.

Edit

I was curious and looking at the code it seems that the problem is in your application.This seems to fix the issue (I can't enter a newline on that box, but I can't say if this is normal or a bug). Those values (100, 200) were selected randomly so this may not be the best way to solve that.