Many years ago I went to a lecture on earthing aerials. It was interesting to say the least. He pointed out all the things we should not do like do not mount on an earthed structure soot lined chimney is out. Also not on sharp edges should not use gable end of house.
It would seem we have been getting it wrong for years but to be fair he was talking about a G5RV on 80 meters not a TV aerial. However some of the points raised are still valid.
The TV aerial is not really a problem as we can use decoupling capacitors however for a dish we need a DC supply. However unlike the Yargi we don't need to mount a dish high up and mounted on the wall unlikely it will get a lighting strike. So 10 foot off the ground can't really see a problem in earthing the dish.
However I question why the dish should make the set top box live? Should be 300mm separation or an earthed divider between the data cable (coax) and any low voltage (230vac) cables. Even if the rules were not followed the chance of getting a shock are very low.
So I would be looking for a fault. First thing I would look for is any combiner/splitter unit likely fitted in loft which may be faulty. Either it could be connecting the aerial to mains supply or connecting the dish to the terrestrial aerial without using a decoupling capacitor.
Next would be earth system in house. Not unknown for back boxes to become live and transfer that to other items.
On change of occupant or every 10 years one is suppose to have an Electrical Installation Condition Report was called a Periodic Inspection Report this should highlight any faults. I have been surprised to find quite so many cases of line - neutral swaps.
Something like
this is not as good and a proper EICR but in real terms is likely to highlight most of the faults. Note I have shown one with a loop test.
I will guess there is a major fault some where. But the chances of working out where on a forum are slim. I would recommend getting an electrician and at £42.30 plus vat for tester likely the electrician is the cheap option.