OK - the extra info helps.
I think you have an older TT installation. This means because your supply comes in on overhead wires, the earth is provided by you - the customer, and not by the electricity company (who only supply live and neutral).
Because customer supplied earths are considered less reliable, slightly more elaborate protection is provided at the consumer unit (hence your
2 ELCB's).However ELCB's are the older way of doing it. Now they use RCD's - which are considered more reliable. (Basically an RCD will work correctly regardless of how good or bad your earth is, whilst an ELCB may not).
One option available is a replacement consumer unit with appropriate RCD's (TT installations usually have 2, one less sensitive one for the whole house, followed by a second more sensitive one just for the ring mains).
Another option is to have the TT supply upgraded to TN-C-S (also known as PME) as well as a simpler single RCD consumer unit (IMO this is the better option).
Either way I would still have your installation checked out by a professional, and your earth tested (check they are able to do this).
Regarding the dodgy socket - it could have all 3 wires transposed (live is connected to neutral, earth to live, and neutral to earth). This would explain why very low powered devices could be plugged in and work ok (because they usually don't pass enough current to trip the ELCB), but anything larger does.
Bear in mind some plug-in socket testers are unable to detect earth/neutral transposition.