In Japan they often use Paulownia wood because it does not have strong growth rings with encourages movement with a change in humidity. This is a pithy type of wood and a similar one here in the west would be Magnolia. Balsa would be the extreme type of pithy wood but the strength levels are low. You need a wood that just strong enough so that it can be shaped but one that will not dramatically shrink or contract too much over the seasons. To adhere the stone to the dai I would just use some type of Latex caulking which bonds well enough while not being so permanent in case you decide to release the base and the stone.