My question is about Zacama, Primal Calamity or any creatures with abilities that are activated using mana. If such a creature is destroyed, can you activate its abilities as an instant after it's been destroyed.
No, you cannot activate an ability once a creature has been destroyed; when a creature is destroyed, it is moved to the graveyard immediately without players getting priority, so there is no chance to activate any abilities, and Comprehensive Rule 113.6 makes it clear that abilities function only on the battlefield (unless some specific exceptions apply):
Abilities of an instant or sorcery spell usually function only while that object is on the stack. Abilities of all other objects usually function only while that object is on the battlefield.
What is possible (and common) is to activate the ability in response to the spell or trigger which would destroy the creature going onto the stack, but before it resolves.
While Philip's answer is basically correct (barring some what might as well be invisible differences to your typical player) I fear there is a misconception here that should be addressed, and is indeed touched on at the end of his answer.
Say for a moment Bob has his Zacama on the battlefield and Alice casts Murder targeting the Zacama. A novice player may throw up their hands and say "Welp, I don't have a counter spell so bye bye Zacama!" when in reality they can choose to activate any activated ability (unless it specifically says it's only at sorcery timing) or cast any instant from their hand.
Imagine you have a stack of notecards with your actions for the game. It starts empty - no notecards. Alice puts her notecard on the stack "Alice casts Murder targeting Zacama". She can actually put as many notecards on the stack with all her different actions. She can choose to activate abilities of her own creatures, tap lands, play instants. When she's satisfied with her actions, the next player (which in a typical game is just the opponent, in this case, Bob) gets to do the same. Bob can activate abilities of his own, again putting his notecard on the stack "Bob activate's Zacama's first ability targeting Alice's Grizzly Bear with 3 damage". When he's done, it goes to the next player, which here is simply Alice again.
This process can repeat many times and, in some multiplayer games (or even high level two-player games) can get quite complicated as players are responding to each other's spells. Normally, it doesn't get too out of hand because most of your abilities have a cost, either a tap or mana, and eventually people run out of those resources and can no longer respond.
When no one chooses to respond to the most recent action by putting their own response action on top, the game takes the top notecard and follows the instructions. Then the process of "do you want to take an action" goes around the table again (starting with the player whose turn it is.) If someone does, great, the process repeats. If it doesn't, the top notecard comes off and is followed by the game.
This concept of little notecards on a stack is actually a fundamental part of how the game works - they even call it "the stack" in the rules. In fact, "Any time you could cast a sorcery" is defined as "any time that stack of notecards is empty and it's your turn".
In case that was all too confusing, the very important thing here is that just because Alice has cast her Murder spell does not mean the creature is destroyed. All players (including Alice) have a chance to respond to Alice casting the Murder spell before it is "resolved".
So while you're asking if the abilities can be activated right after it dies, the answer is no, no it cannot. But the real question is can you activate the abilities right after someone tries to kill it, and the answer is yes, yes you can - as soon as they are done activating their instants you can respond.
(And, I'll add that it is almost, almost always only one spell or ability going on the stack. Very very few times to people respond to their own spells or abilities, because if you're going to do something to screw up their plans they would prefer to not waste their resources on two parts of their combo before you do - they'd rather give you a chance to respond and if you're going to foil their plans you do and then they proceed to the second part of their plan.)