We used to have a BBQ skewer that we used for various purposes, including adding holes to belts. We'd heat it up on the gas hob and then burn a hole through the conker. I actually still have the same one I've inherited in my kitchen drawer. If you have an awl, you could use that instead, but I'd recommend heating it to get a cleaner hole.

You need to use a long enough string. Old cotton shoe laces are actually perfect as the aglets make threading that much easier.

The force of one conker against another is enough to sometimes make it spin round, but not enough to do any real damage. You just need a long enough string that your fingers are not in the firing range. Obviously there is a vanishingly small risk of a piece of conker ending up in the eye but I never witnessed that or any other injury happening. The biggest problem was usually upset kids when their prized conker got destroyed.