Okay, I'll break this apart. Theravadin Buddhism teaches that when you die, your consciousness gets reborn in another body. And this consciousness is affected by your Kamma in a way that everything you've done in the past, and said in the past, and thought in the past, is your Kammic make-up that attaches itself to another body which is appropriate for what you've done - whether it's a lot of good Kamma or not a lot of good Kamma. So you get instantly reborn into another body.
Now, why is it important what we think before we die? Because, according to Buddhism, you have two factors that are going to push you into a new body. This is what's taught. One factor is your basic Kamma - everything you've said and done and thought - how good of a person you were. If you were a very good person then you will be pushed by your Kamma to get a good rebirth; you will have a favorable rebirth in a human realm, or maybe in heaven realms. So that's your good Kamma, it will push you in that direction if it's a lot. Of course if it's a lot less than your bad Kamma, the result will be vice versa.
Now besides the Kamma pushing you, you have a second factor which can also push you. Even if you have done lots of good Kamma, if you have some unpleasant, unwise, maybe angry thoughts, at the moment that you die, the anger is what will push you then. So you're not going to get a good rebirth even though you have lots of good Kamma doing the bigger push. That lesser little push will push you in the direction of a lesser rebirth. Now if you have tons of good Kamma and you think angry thoughts when you die, according to Buddhism you will get reborn according to the anger, but because you have lots of good Kamma, that will only be a short life, then you will be reborn again into a better place where your good Kamma fruits. This is what they teach. So that little thought at the moment of death, if you can make sure it's not a negative or defiled thought, then your good Kamma will continue to push you to your next life without any problems?.
Now the question: Why is giving ones organs after death not permitted? I don't know that it's not permitted. It probably wasn't a question back in the Buddha's days, so he probably never said anything about it, so I don't know that this is not permitted. It may be that you don't want to do it, I personally don't want to do it, but I don't know anything about it being permitted or not permitted.
What happens for example for people who die during extreme sports at the moment of highest excitement or concentration? I can't tell you exactly; nothing I've read explains this particular question. All I can still assume is that, the moment of death is still exactly the same. Your last thought at the moment of death is going to push you one way or another, so if your super excited in a negative way it will push you towards the negative Kamma coming forward; and if your super excited but yet still in a positive way, then I assume the good Kamma still takes over, but specifically I can't really answer that question.