Yes! Nigel is correct... 0.4mm (15thou)...
No idea why it's stuck in my mind - other than an embarrassing incident 40+ years ago - continued below!.
If its a wet damp morning and the plug leads are worn out, ie insulation is poor.. Then a drenching of WD40 will get you going... It seems that the plug leads are tired....
Damp and wet - I was out in the car with my girl friend, and we came to a ford (there aren't many round here, but I'd been through it many times - including on push bikes). The water looked fairly high, and there was a car sat waiting - too scared to go in - so I went past him and cautiously entered the water. We got half way across and the engine stopped dead - and wouldn't attempt to restart.
'Luckily' we'd been out walking, and had wellington boots with us - 'unluckily' we'd placed the boots in the boot (where else would you put boots?). Anyway, I'd just got out and was wading through the water when a Subaru pickup comes flying down the hill, he spun round, reversed towards me, and threw me a tow rope - I tied it to the car and he pulled me out. He then proceeded to drive through the ford at about 40mph, causing a massive wave - no one likes a show off!
I opened the bonnet, and tried to dry the electrics as well as I could, but it didn't look wet at all, and it made no difference - now if you remember the start of this story, the guy too scared to go in? - he came across with a can of duck oil (like WD40 but in a pump spray can), we gave it a couple of sprays and it burst in to life - missing a bit at first, but soon settled down to run normally.
Needless to say, I chickened out, and didn't attempt the ford again - so turned round and went back the long way - driving with no shoes, socks, or trousers. We got back to my parents house, walked in (minus trousers) and sat down on the sofa watching TV - it was ten minutes before anyone noticed I wasn't wearing any trousers!.
The reason for this rambling story, and perhaps why I remember the 15 thou, is that the problem that caused my engine to stop was the points - the gap was too small, and the the spark too weak, so a little dampness stopped it working at all.
For the OP's problems - the assumed 'shorts' to ground are red herrings, it's simply low resistance electrical items across the supply, perfectly normal in a car. My main concern would be the worn wires, and particularly where some are showing bare copper - it sounds like the wiring loom may need changing?, or at least parts of it. A quick google shows that wiring looms are easily available.