Is it possible to add volume controls & shortcut keys to a mechanical keyboard which does not have the keys?

You could use a generic hotkey utility like AutoHotkey to assign keyboard shortcuts (something unlikely to be used by normal applications, like Ctrl+Alt+F12) to perform your desired special functions.

Examples:

Windows+F11 increase volume:

#F11::Send {Volume_Up}

Ctrl+Alt+Numpad0 start Calculator:

^!Numpad0::Run Calc

See the linked pages for hotkey reference, changing volume.

I found some interesting things in researching this, but this is probably the most interesting:

http://www.piengineering.com/xkeys/xkstick.php

The X-Keys Stick is very spendy at $99 though.

The other alternatives are things like an inexpensive USB keypad, which can be had for under $20, and you could remap its keys via software to multimedia functions. I did find one with dedicated multimedia keys, but it's... odd. The model # is 1000-BRF.

Beyond that, gaming related devices like the Cyber Snipa game pad, or the Logitech G13 gameboard.

More obscure stuff would be shuttle controls like the ShuttleXpress or the Shuttle Pro