I'm curious, when there is a character effect on the chain, such as Destiny's ability, would it be possible to negate it w/ Sentinel Mark II's ability (Activate, discard a Sentinel character card from your hand >>> Negate target activated character effect.) while it's still waiting to resolve?

As in, my opponent has initiative and recruits a Destiny, then he uses Destiny's ability (Activate >>> Whenever target character becomes stunned this turn, its controller loses 4 endurance. Use this power only during the build phase.) before I'm able to recruit a Mark II. Would I be able to negate it once I recruit my Mark II that turn?

Sorry, but by the time you recruit your Mark II, Destiny's ability would have already resolved. You would only be able to stop it if your Mark II was already in play.

But the Sentinel doesn't say anything about negating an unresolved activated ability. How can you negate something that doesn't technically exist?
I have a similar question regarding Not So Fast. If you play it on the chain before the plot twist has resolved, how can you negate it? But if you have to play it after the plot twist has resolved, if it was something like Reconnaissance, it's a little too late, isn't it?, or do you have a response window between the time the plot twist resolves and when it takes effect, allowing you to put Not So Fast on the chain. This doesn't seem to jibe with everything else that I have learned, so I wait for someone official to come along.
Until then, the Sentinel can negate any activated ability at any time, and Not So Fast has to be played after the fact, and the occassional card that comes along that presents a problem is just the exception that makes the rule.

Effects exist on the chain.
Non on-going plot twists generate effects.
Recruting a character/piece of equipment is an effect.
Payment Powers create effects.
Most triggered powers when their conditions are true, create an effect.

Surfer 13 said:
"Until then, the Sentinel can negate any activated ability at any time"

No, it has to negate it at the time of activation, when the ability is on the chain before it resolves. It cannot Negate a resolved ability.

i.e.
I activate my Destiny and her ability goes on the chain, I pass Priority to you and you have to activate Mark II's ability NOW to negate Destiny's power, putting his ability on the chain. You pass P, I pass P and the chain resolves in Last in First out order, your Mark II negates Destiny's ability first, and Destiny's ability fails to resolve because of that negation.

If You wait to do it later when your chance to get in on that chain is gone, Destiny's abilty has already resolved and you can't negate it.

what if destiny was stunned? does the effect still takes place?

Stunning her in no way negates her abilty. Once Her ability resolves, it's there no matter what happens to her.

So you have to use MarkII and Not So Fast after the power/plot twist has been placed on the chain, but before it resolves?
Like I said, it seems strange that you can negate something that doesn't exist, but this is also what my gut told me. I guess if this is the case, then that will teach me to follow my head instead of my heart.

Just because the effect on the chain hasn't resolved yet, it still exists, for the game mechanic's need to be able to reinact what would happen in the Comic world. Counter measures (like Counterspells in Magic) need a moment to respond to an effect and counter it.

To use Danny Mandel's description of comic powers in the game world (in his Top Down design Column, on Metagame.com):

Mark II see's Destiny "Powering Up" to make her prediction, and reacts with his own "Negation Bolt" shooting from his palm and negating her power for the time being.

mark II's ability can be used outside of a chain, so after destiny's ability is used
if you can recruit a mark II in you build phase than you can negate destiny's ability

No, you can't... read the previous posts again as to why.

Apply it to different character with a different abilty. I use Mr. Fantastic to look at the top four cards and put an equipment in to my hand during my build phase. it resolves, and we go to your build phase. you recruit Mark II and try to negate Mr. Fantastic's ability, but you can't, because it has already resolved and I already have the card.

It's the same thing with Destiny, as long as her ability has resolved, it's too late to back up and negate it. Just becuse her ability becomes a triggered affect that sticks around untill the predicted creature is stunned, doesn't mean you can negate it later on, it's too late, the milk is spilt and there's nothing you can do about it.