Unfortunately, you can't have Mathematica keyboard shortcuts work through the browser, since the browser is likely to use the keyboard shortcuts itself and each browser will use them in different ways.
And, as far as I know, merging cells through the menu items is just as easy as copying and pasting the contents of cells.
If you think having this as a menu item would be useful in the cloud version, try filing a suggestion with the Mathematica online developement team thru the "feedback" button at the bottom of the page.
