Your car may be too old to have the in-line fuse behind the panel.
I think I already mentioned this, but JIC : the fuse ratings given in the Triumph documentation are using an obsolete Lucas rating method, that results in the numbers being about twice what a modern fuse of the same capacity would be. Thus if you are going to use modern fuses, they should be rated 20 and 25 amps respectively.
There was an interesting demonstration at VTR/SP a few years back, where they first loaded a proper Lucas fuse until it blew, and then inserted an AGC fuse with the same rating, plus added some more load. Not only did the fuse not blow, but a minute or so later the smoke started leaking out! There have been cases reported of the same thing happening with people's cars ...
From your description, I strongly doubt that all 3 symptoms have the same cause. For the panel lights, are you sure the switch is turned on ? (Both the headlight switch and the 'panel' switch must be on for the panel lights to come on.) Do the lights in the tach and speedo work? If all 6 bulbs (2 each tach/speedo, 2 more for other 4 gauges) are no-shows, but the taillights work, I would guess it's the 'panel's switch or the wiring to it.
For the 'high' horn, it might be the wiring to the horn (I fought an intermittant horn problem for a long time that turned out to be a connector sleeve that had broken internally and wasn't making good connection). The horn don't depend on a ground to the body, they get grounded through the horn button when it's pressed.
But more likely it's the horn itself. There is a pushrod inside that operates the contacts, and it tends to get corroded and stuck. The contacts could probably also use cleaning and adjustment.
Heater fan could be either the control (they are notoriously troublesome) or the fan motor itself. There is a separate ground wire, so not likely to be a ground problem unless someone has left the wire disconnected. Or perhaps the power wire is disconnected or broken. This is one of those cases where it would make a lot more sense to poke around with a test light, before starting to replace things. Ground one side of the light and turn the key on before you climb under the panel, then touch the probe to each terminal in turn of the heater control. No lights means the wire from the ignition switch is bad; one light means the control is bad; two lights means the problem is the motor (or wires from it to control & ground).