Tiger 800 Questions

Hi and welcome to the forum, a great choice of bike!
I would assume that the fog lights are the LED type, (the early ones had halogen bulbs) fitted to the later models. If so it appears that you have to replace the complete light unit, go onto www.worldoftriumph.co.uk or www.fowlersparts.co.uk. The part number you need looks as if it is either T2701465 or T2706027 depending on the model year you have, a warning however the price of a light unit is just over £140!! I suggest you check the wiring first to make sure that it is the light unit that is defective.

Your second query regarding the powered top box, this looks like a home made addition, the genuine Triumph one had contacts under the top box and a socket on the plate so that it disconnected when you took the top box off.
You need to check whether your leads are powered all the time or just ignition live and you need to check the feed and fuse rating to see what load it can carry.