Assuming you are an active player you play an Oracle of Mul Daya or Crucible of Worlds. Could you play your land before they kill Oracle of Mul Daya? Would you be able to play lands off of Crucible of Worlds if they would try to destroy it?
APNAP doesn't factor in. APNAP is used when multiple players need to make choices simultaneously, and when stacking triggered abilities of multiple players. These questions are strictly about priority.
As the active player, you get priority after a spell resolves.
116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
If you were to play a land then (a special action), you would get priority again.
116.3c If a player has priority when he or she casts a spell, activates an ability, or takes a special action, that player receives priority afterward.
So the answers to both questions are yes.
The other answers are partly correct. The active player gets priority after a spell is cast or an ability activated, and playing a land is a special action that does not use the stack. If a spell is cast that would destroy a permanent that allows you to play a land in a different way, you would not be able to play a land before it resolves. Rule 505.5b explains that you cannot:
505.5b During either main phase, the active player may play one land card from his or her hand if the stack is empty, if the player has priority, and if he or she hasn’t played a land this turn (unless an effect states the player may play additional lands). This action doesn’t use the stack. Neither the land nor the action of playing the land is a spell or ability, so it can’t be countered, and players can’t respond to it with instants or activated abilities. (See rule 305, “Lands.”)
Section 305, mentioned there, says the same thing:
305.1. A player who has priority may play a land card from his or her hand during a main phase of his or her turn when the stack is empty. Playing a land is a special action; it doesn’t use the stack (see rule 115). Rather, the player simply puts the land onto the battlefield. Since the land doesn’t go on the stack, it is never a spell, and players can’t respond to it with instants or activated abilities.
This rule mentions section 115, which also says the same thing:
115.2a Playing a land is a special action. To play a land, a player puts that land onto the battlefield from the zone it was in (usually that player’s hand). By default, a player can take this action only once during each of his or her turns. A player can take this action any time he or she has priority and the stack is empty during a main phase of his or her turn. See rule 305, “Lands.”
(all emphasis added)
It works the exact same way as the planeswalker case, Mentioned in your other question
Once Oracle resolves, assuming that it's your turn, you have priority, and you have all day to take whatever actions are legal to you, until you pass priority. This includes playing your second land for turn(which doesn't use the stack anyway), and you'll still have priority after that.
For instance, if you were to play Fastbond, you could play any number of lands, without your opponent receiving priority, assuming that no triggers are being put onto the stack.