Estimating population mean $\mu$ based on a sampling distribution I've learned that under certain codnitions I can assume the mean of the distribution of sample means to be approx. equal to the real mean of the underlying population. Additionaly, the standard deviation of this distribution tells me how close the sample means are to the real mean. 
This leads me to the following questions:
1.) Can I take just a single sample, calculate its mean, and assume it to be approx. like the population mean?
2.) In order to calculate the standard error, I need the population's standard deviation. How do I know which value it has, as I'm not able to observe the population?
3.) If I take a sample and calculate its mean, I can calculate the likelihood of observing this value using the z-score. However, to calculate the z-score I need the real mean of the underlying population.
 A: *

*If we assume that a population mean value exists (which would not be the case for e.g. the Cauchy distribution), the mean value of the sample $\bar{y}$ is always an unbiased estimator for the population mean value $E[y]=\mu$. Thus, the answer to your first question is yes! However, this statement does not quantify the quality of the estimate.

*In order to quantify the quality of the estimator in (1) we "usually" use the standard deviation of the sample, $s=\sqrt{\hat{Var}[y]}$. The sample variance $s^2$ is an unbiased estimator of the population variance $\sigma^2$, if we assumes independence of the data points. By using the central limit theorem, we can estimate the quality of the estimator in (1). However, since we do not know the population variance but only its estimate, we do not use the Normal but the $t$-distribution. The $t$-distribution considers the uncertainty related to the estimate of the population variance.

*Probably the best would be to calculate the $t$-value and not the $z$-value. However, even the $t$-distribution does not consider the uncertainty of the estimate of the population mean. Thus, you have the same problem: Formally you need the population mean, but you only have the sample mean. Nevertheless, the uncertainty associated with the estimate of the population mean (=sample mean) is usually smaller than the uncertainty associated with the estimate of population variance (=sample variance). Hence, we usually take the $t$-value despite this formal shortcoming. 
