Is it a must that the population sample has to be normally distributed in order to work with student T distribution and obtain confidence intervals based on that? As I know, this is not a must for the 1st case (i.e. when we have the population mean). Is it a must for the second case though?

I think if we can assume the CLT applies then we can assume normallity of the underlying distribution. In this case we are targetting the 'sampling' distribution of a sample statistic in question which doesn't matter what the underlying population distribution is. If the sample size is too small and we cannot assume CLT then unless the underlying distribution is normal we cannot use the T distribution.

No, but very importantly we do assume the population data from which we drew our sample to be normally distributed. In the formula we use the sample mean (x-bar) which is an estimate of the population mean (mu) and not the population itself.