The nasty sticky sap is exclusively in the bark. It's best to remove the bark (wearing gloves) before doing anything else. Like most coniferous woods, monkey puzzle is not prone to cracking/checking. Compared to pine or spruce it is relatively hard, but not as hard as yew. There are some silicates in the wood, and they do have a distinct dulling effect on your tools, but overall it is of very even consistency.
The big exception are the knots. Monkey puzzle grows its branches at regular intervals along the trunk, and the branches form very even circles. This can be used to spectacular effect on end grain bowls or platters, but watch out: the branch knots are very dense and hard, much harder than the rest.
You turn it wet or dry, and there is very little movement in the wood. Pete is right about the pith: it is very soft and should either be avoided, soaked with CA glue or replaced.