I will answer your point about simulations with R because this is the only one I am familiar with. R has a lot of builtin distributions which you can simulate. The logics of naming is that to simulate a distribution called
dis the name will be
Below are the ones I use most often
# Some continuous distributions.
# Some discrete distributions.
You can find some complements in Fitting distributions with R.
Addition: thanks to @jthetzel for providing a link with a comprehensive list of distributions and the packages they belong to.
But wait, there's more: OK, following @whuber's comment I'll try to address the other points. Regarding point 1, I never go by a goodness-of-fit approach. Instead I always think about the origin of the signal, like what causes the phenomenon, is there some natural symmetries in what produces it etc. You need several book's chapters to cover it so I'll just give two examples.
If the data are counts and there is no upper limit, I try a Poisson. Poisson variables can be interpreted as the counts of successive independent during a time window, which is a very general framework. I fit the distribution and see (often visually) whether the variance is well described. Quite often, the variance of the sample is much higher, in which case I use a Negative Binomial. Negative Binomial can be interpreted as a mix of Poisson with different variables, which is even more general, so this usually fits very well to the sample.
If I think that the data is symmetric around the mean, i.e. that deviations are equally likely to be positive or negative, I try to fit a Gaussian. I then check (again visually) whether there is a lot of outliers, i.e data points very far away from the mean. If there are, I use a Student's t instead. The Student's t distribution can be interpreted as a mixture of Gaussian with different variances, which is again very general.
In those examples, when I say visually, I mean that I use a Q-Q plot
Point 3, also deserves several book's chapters. The effects of using a distribution instead of another are limitless. So instead of going through it all, I will continue the two examples above.
In my early days, I did not know that Negative Binomial can have a meaningful interpretation so I used Poisson all the time (because I like to be able to interpret the parameters in human terms). Very often, when you use a Poisson, you fit the mean nicely, but you underestimate the variance. This means that you are unable to reproduce extreme values of your sample and you will consider such values as outliers (data points that do not have the same distribution as the other points) while they are actually not.
Again in my early days, I did not know that Student's t also has a meaningful interpretation and I would use the Gaussian all the time. A similar thing happened. I would fit the mean and the variance well, but I would still not capture the outliers because almost all data points are supposed to be within 3 standard deviations of the mean. The same thing happened, I concluded that some points were "extraordinary", while actually they were not.