Is it possible to make the non-separable data more separable by any methods of feature selection, extraction or transformation? Could these data (in the figure below) be separated by any means of feature extraction, transformation, or it's just a waste of time to make the three classes separable if they "in fact" weren't separable at all?
I have used Mutual Information, PCA, and Isomap as feature selection techniques and this is the best that I got of data segregation! 

 A: If the process that generates labels for your data is not random we may assume that there exist an algorithm that (given data) can separate the data better than random guessing.
The case may be that the data you have may not be sufficient to explain your independent variable - predict your class labels.
As you have mention, some machine learning algorithms (especially when you have lots of data) can benefit from doing feature extraction.
What I would do, assuming that you have data labels available prior modelling, is applying an machine learning algorithm (such as random forest) to see if it can be used to fit the data.
A: Clearly, feature selection and linear transformations (including PCA) cannot help you there. If the data is not separable in $n$ dimensions, it will remain inseparable in $m<n$ dimensions. Imagine projecting your data onto any 1-D line in that space. 
Non-linear transformations may help you, and in theory Isomap is a non-linear transformation, but my intuition tells me that Isomap should not be of any use to separate the data. The standard isomap is "color blind" or "class blind" - it will find the "curved subspace" (or the manifold) where most of your data lies, and as far as I can see that will not separate your classes, unless your data was already "separated" but "not linearly" See for instance the figure on the Swiss roll here https://www.researchgate.net/figure/11580034_fig1_Fig-1-A-The-Swiss-roll-data-used-by-Tenenbaum-et-al-1-to-illustrate-their  The classes/colors in figure A are "separated" and the isomap just flattens the subspace (figure b) and now the data is linearly separated.
