Take a look at the photo below. It's an antique wet cell used in automobiles way back in the day. As you can see, they were often made of glass with a wooden basket keeping the cells together.
If you set such a battery on metal or concrete, you risked two potential hazards: cracking the cells on impact and swelling of the wood as it absorbed moisture from the floor, which could again lead to cracking. Setting any battery on the floor also makes it a risk for short-circuiting if one should drop a wrench or other metal object on top of it.
Modern batteries are made of plastic, and as long as you don't drop them and crack the case you can sit them on any material you like with impunity. There is absolutely no way that the ions in a battery can travel out of the cells without an electrical path between the poles.
With the old carbon-core batteries, sitting on concrete was an issue, would discharge to ground. Nowadays, no such issue.
Yes, if the metal table is sitting on the ground or wet concrete it will. You should always set a battery on a piece of wood to isolate it from the ground. The acid residue is conductive and free ions in the battery will migrate to earth ground potential.
Nope! It will be just fine!
No. The battery tray in a car is usually made of metal.
