Do Events Take Place Inside Black Holes? As you approach a black hole, the universe you observe slows down. When you see someone from outside approaching a black hole, you see how they freeze in time as they get closer and closer to the event horizon. Their frame of reference slows down as seen from the outside. In fact, all of time passes for the observer far away from the black hole, before the other observer reaches the black hole ... or at least that's what I take away from it. (But tell me, please: Does this period of observation of everything (if this is the case at all) last for a long period of time (I'm thinking about exceeding the life span of a human) or (what I think is more likely to be true, from my "concept" of it) a very short period of time (a fraction of a second from some point in the future to all eternity)?)
Yet we know that black holes don't last for all eternity. They radiate away. Very slowly when they are big, but they do disintegrate and finally vanish in a finite amount of time.
If, before a black hole can be reached by things falling in, all of time passes (when observing the rest of the universe) and after some finite time there is no black hole any more, does this mean that no event can take place inside a black hole?
As you probably have figured out, I'm assuming that from the fact that time for the rest of the universe speeds up as you approach a black hole, it follows that an observer on the inside of the black hole experiences no time before either all of time has passed for the rest of the universe or until there is no more black hole. While I can imagine this to be true I never heard of this and can come up with a different hypothesis. However, this one seems more likely to me. Also, the other one would answer "no" to my question, so there's that. 
 A: Of course time to collision is short, plus tide effects might tear matter appart.
But "beside", in the local frame of the observer, nothing change: events still happen.
Anyway, take care about what you exactly call "black hole" ( Schwarzschild limit vs the central singularity in RG model alone  vs the real-world "central thing").
If it is the Schwarzschild limit, then for giga black holes like the one at galaxy center nothing special happens for you at this limit (e.g., no significant tide effects), and you are still very far to the collision. So normal events still do occur for you.
A: 
Do Events Take Place Inside Black Holes?

Opinions vary. If you have a look at the Mathpages formation and growth of black holes you can read about two interpretations:
"Historically the two most common conceptual models for general relativity have been the "geometric interpretation" (as originally conceived by Einstein) and the "field interpretation" (patterned after the quantum field theories of the other fundamental interactions). These two views are operationally equivalent outside event horizons, but they tend to lead to different conceptions of the limit of gravitational collapse. According to the field interpretation, a clock runs increasingly slowly as it approaches the event horizon (due to the strength of the field), and the natural "limit" of this process is that the clock asymptotically approaches "full stop" (i.e., running at a rate of zero). It continues to exist for the rest of time, but it's "frozen" due to the strength of the gravitational field. Within this conceptual framework there's nothing more to be said about the clock's existence."
Most people only know of the other interpretation, where the infalling body goes through the event horizon. From what I know of general relativity, I'd say that's wrong, and that the correct interpretation is the one where everything stops at the event horizon.

As you approach a black hole, the universe you observe slows down. When you see someone from outside approaching a black hole, you see how they freeze in time as they get closer and closer to the event horizon. Their frame of reference slows down as seen from the outside. In fact, all of time passes for the observer far away from the black hole, before the other observer reaches the black hole ... or at least that's what I take away from it. 

Me too. Gravitational time dilation goes infinite. Many people will tell you the Schwarzschild event-horizon singularity is a mere artefact, and that you can adopt another coordinate system to do away with it. I disagree with that, and say that Kruskal-Szekeres coordinates commit the schoolboy error of putting a stopped observer in front of a stopped clock and claiming that he sees it ticking normally.  

Yet we know that black holes don't last for all eternity. They radiate away.

No, we don't know that. Hawking radiation remains hypothetical. Moreover the "given" explanation has particles popping into existence, negative energy particles, and no gravitational time dilation. I think it's unsatisfactory. 

If, before a black hole can be reached by things falling in, all of time passes (when observing the rest of the universe) and after some finite time there is no black hole any more, does this mean that no event can take place inside a black hole?

I think no event can occur because its a place where gravitational time dilation is infinite and the "coordinate" speed of light is zero. Not because the black hole somehow disappears.  

It follows that an observer on the inside of the black hole experiences no time before either all of time has passed for the rest of the universe or until there is no more black hole. 

Agreed. 

While I can imagine this to be true I never heard of this

Google on Oppenheimer frozen star and you can read about it. I'm afraid what sometimes happens in physics is that the people who promote a particular interpretation want you to think its the only interpretation, and that what they say cannot be challenged.   
