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.

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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

Assuming you are using Windows, and if you can do without program shortcuts (I just use WinKey search), you may find 3RVX useful. You can use it to add keyboard shortcuts for volume control as well as mouse combinations. I use WinKey + mousewheel to adjust volume. It also adds an on screen display volume slider. It works well, but as it runs on .NET, it does use more memory than it should for something that runs all the time. The author is making a new version which does not run through .NET though. (I don't know what happened regarding that - there's no longer any mention of it)