While "while" can be used to indicate synchronicity and contrast, "whereas" can only be used for the latter. Thus, your last example would have to be something like: "I can stay at home and listen to podcasts, whereas you have to go to work and make money" - "I X whereas I Y" sounds (logically) wrong to me.
"while" can be used for synchronicity and by extension for contrast (so both "I can listen to English podcasts while I am going to work" and "I can listen to podcasts while you cannot" are OK). Actually, there is a slight difference between "while" and whereas even when used for contrast. "While" really means at the same time, so you can also use it for mild contrasts like "while I don't care too much for podcasts, I listen to them once in a while" (whereas does not fit here). "Whereas" is closer to "on the other hand": "You love podcasts whereas I don't." (while would be OK here, too). So, they can be synonyms in some cases, but not always. When in doubt, use while ;-).
"whilst" is indeed a synonym of "while" - it sounds rather archaic / literary to me, but that may also be a regional question ...