Some fag-packet maths says that limiting it to 6A and only charging at times when we're at home and awake, will add more miles than we use (over 250 miles/week added vs under 200 miles/week used)... so if we do use granny charging as a stop-gap until the proper charger fitted (I'm expecting a quote for that today) we should be able to do so with safe-ish limits.
My gut feel is that 8A would be a good compromise for granny charging from a non 'EV' rated socket. If the plug or socket get significantly hot with an 8A load then they should be replaced - there is a clear fault with them.
Charging at 6A is, obviously, safer still. But bear in mind that there is likely to be 200W or more of quiescent vehicle electronics running when charging, so around 1A of your charge rate will be "wasted" powering the vehicle. This is in addition to the 10 to 15% inverter conversion loss from AC to DC. Hence at 6A, you will be only be putting in around 5A into the inverter (hence a loss of 16% on top of the inverter's losses).
FWIW... The power dissipated at any point of resistance (assuming a largely resistive load) is proportion to the square of the current drawn. From Ohm and Watts laws (V=IR and P=VI) you get P=I * I * R.
Hence, if you dodgy plug and socket has a resistance of (say) 150 mOhms, then the power dissipated by that connection would be:-
At 6A; power = 5.4W
At 8A; power = 9.6W
At 10A; power = 15W
At 13A; power = 25.3W
Yes. Not sure how it was used, but it is never a good idea to force disconnect an inductive load through equipment that is not designed for it (such as a regular BS1363 plug/socket).
If one is using a granny charger, I'd always ensure that the charging has been terminated from the vehicle before switching off or unplugging a 13A plug - they are not designed for interrupting inductive loads.