I would definitely suggest seeing if the window manager is configurable in this regard to ignore those keys if it recognizes them, since that's generally the reason why keys don't appear in the Sublime input log.
That said, on my Linux machine only the Mail/Explorer/Calculator buttons are handled by the WM but the others are not (This keyboard doesn't have a key for Web, but it's similarly mapped by the WM). I presume that's because the dedicated application buttons are more global than say back and forward, which is better used as an application specific key binding.
Generally speaking, if pressing the key while
xev is running gets it to generate output, the WM is probably not catching it;
xev won't tell me the key code for XF86Mail unless I tell the WM to ignore it.
Something to try would be to add the following lines to
~/.Xmodmap (or create the file if it doesn't already exist):
keycode 166 = F13 NoSymbol F13
keycode 167 = F14 NoSymbol F14
Once you do that, execute
xmodmap ~/.Xmodmap in a terminal to make the bindings active right away.
Here the 166 and the 167 are the key codes from
xev for the
XF86Forward keys; the part after the
= remaps those keys to be F13 and F14 respectively. If you use input logging in Sublime as you outlined above, you should see those keys being reported in the console when you press them, and they should then be bind-able as well.
You can extrapolate bindings for the other keys as needed, although you may need to tell the WM to keep it's fingers off of the application keys if you want to use those for something. There may well be better substitute key mappings than these (and it may not be a good idea if your keyboard has a physical F13 key ) depending on what you're trying to do.