There's no big difference between "empezar a, ponerse a, y echarse a". I find them similar to english "start to". In Argentina "echarse a" is not as common as the other two, so I think the frequency of usage depends on localization. But, no matter where you are, you can say for example "empezar a jugar", "ponerse a jugar", "echarse a jugar" (the last one sounds odd to me, because echarse also means "to lie down", which changes the meaning of the phrase). The verbs that come to my mind, can all work the same way with the 3 options: caminar, beber, andar, estudiar, pintar, etc.
As for estar, seguir and llevar, there is a little difference. The last one, "Llevar + gerundio" gives me the idea that the action has started long ago. For example, with the verb "beber": estar bebiendo means that you're drinking, seguir bebiendo means that you started drinking before and you continue doing it, and llevar bebiendo might mean a longer drinking activity. I'd translate them to english as "being drinking", "keep on drinking" and "keep drinking [for long]".