Does a PC have to be mounted in metal case

To pass the requirements of FCC part 15 class b most computers must be in some sort of conductive case. You will notice that most metal cases don't have any slots or holes larger than about half a centimeter, this has to to with antenna theory.

If you are building the projects for yourself, you do not have to comply to FCC part 15, but if you intend to sell them that is a different story. If you want to build a case out of wood and sell the resultant product (not as a kit, there is a whole weird legal thing about kits) you will almost certainly have to coat the inside with some sort of conductor.

A computer case does not need to be made of metal. The reason most cases are metal is because they conduct heat (helps disposing of internal heat a bit better if there are few fans inside), and more importantly, its cheaper to produce and the end product is less heavy. A computer case is not being used as ground, simply because the case itself does not touch the floor. They usually have rubber feet.

The case may be connected to the grounds of the motherboard, but that is not necessary, and it is mostly done in case the computer is not connected to a grounded wall-outlet.

That said, I've seen many people build computers from various materials, wood included but not limited to. Test pc's often have no case at all, but the best example is the desk pc, a pc build directly into a desk. These desk pc's are often made of wood.