1. Laplace distribution (occasionally double exponential, but both terms are overloaded 2. I don't think people can tell you which metric is best for you unless you give them enough information about what it needs to do that they can make suggestions (though maximum likelihood may be one you won't necessarily think about). You can ask about properties of metrics, though. 3. not sure what you mean by 'normal plots' or 'natural numbers' in this context. Do you mean QQ plots?

4. A histogram isn't necessarily a good choice for selecting between distributions, especially if you only use a few bins

5. Asking about a good way to estimate the parameters of a Laplace distribution can be done on the main site; maximum likelihood will be the almost universal suggestion (median for the location parameter, mean absolute deviation from the median for the scale parameter)