How to fetch submenu items for current page?
You are right it's a pretty common requirement. If you are not afraid of slight performance hit, you may use Menu Block module. It is able to provide you with submenu relative to your current path:

So if you’re only using your theme’s Main menu links feature, you can add and configure a “Main menu (levels 2+)” block. That block would appear once you were on one of the Main menu’s pages and would show the menu tree for the 2nd level (and deeper) of your Main menu and would expand as you traversed down the tree. You can also limit the depth of the menu’s tree (e.g. “Main menu (levels 2-3)”) and/or expand all the child sub-menus (e.g. “Main menu (expanded levels 2+)”).
Pretty simple, eh?

Of course it is a pretty big module, and it puts additional block layout between your menu and your display, so if it is too much of a performance hit, you can try to borrow some of it's code - it's GPL after all.