Hopefully, you will have lots of live kids, and the problem will not arise very often, if at all. But you are being sensible thinking about things - yes, it does happen. In 35 years of goat keeping I have never lost a kid - until this year, when it happened twice in a week.
As to the disposal of a dead kid - well, technically you are not supposed to bury an animal on your land. Before the foot and mouth and its problems came in, I buried all my goats at home when they died of old age - as well as dogs, cats, rabbits etc. I now pay £15 when I take them to the slaughter house. But .....a new born kid, or lamb or whatever will not have an ear tag, so who is to know it ever existed, if you get my drift. I know quite a few horses buried in fields round here, and so long as its well away from a water course, that seems ok.