The continuous distributions are ones that take an uncountable (infinite) number of values within a given fixed interval and a discrete distribution takes a finite number of values in a fixed interval.
The population distribution is basically the limit as if you had infinitely many samples and calculating its distribution and there are different schools of thought on this, but if you are think of an empirical distribution (i.e one based on data), then the population can be considered as if you have an infinite sample size.
With regards to your question on the population mean, the answer is that it's different for every case.
Sometimes we know (or assume) the underlying distribution (parametric) and sometimes we don't (non-parametric). Sometimes we assume we know some parameters for the population, and sometimes we estimate them which includes all possible combinations of parameters (we may assume we know population variance, but not the mean or vice-versa or know none at all but assume normal).
Statistics is mainly concerned with trying to figure out population values given a sample and if we assume population values then we use them but if not we estimate them.
Sometimes we know (or assume) one parameter and don't know another and the point of doing exercises like this is is that when you have to estimate attributes (this is what statistics is all about) you need to know how to deal with the various kinds of information and how they are used to get the most accurate representation for the stuff you don't actually know (or are trying to estimate).