Poisson and Approximating Binomial

[RocketManJames]

Can someone help me understand this a bit, and tell me if this makes sense at all. Years ago, when I took a probability class, we went over the Poisson distribution, which could be used to approximate the binomial distribution for rare events.

So, say that I've got a fairly rare event (probability of this event occuring is about 1 in 100,000 trials). And, say that I've estimated the probability of this rare event from running 30 million trials and seeing it 300 times.

Now, I have a few questions...

1) Can I use the Poisson distribution to approximate the distribution that I would expect to see if I were to run a large number of trials (N = Large)?

2) Is the reason for using the Poisson Distribution as an approximation, because it is simpler (fewer terms, etc) than the Binomial?

3) Is it possible given the information I gave above to provide some range of error for any estimated distribution? Since P(rare event) was estimated from running 30MM trials and seeing 300 occurrences, can we somehow infer some cloud of error around any distribution we come up with?

I apologize if what I am asking is confusing or if it is way off. I'm just trying to learn here.

Thanks.

-RMJ