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take care
Can 'take care' be used when somebody stumbles? It seems to me the proper phrase here would be 'be careful'? And 'take care' separately is used only when saying good bye.... Am I right here? (I am interested only in this phrase taken separately, i.e. 'take care'..... i understand the other usage can be 'to take care of something, or somebody' which is different in meaning).
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Насколько я знаю, take care это то же, что и bye-bye.
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In the sense you're thinking of it literally means "Следи за собой" but you wouldn't use it in this context. If someone falls down or something like this where in Russian you can say like "Тихо тихо!!!" In English you could say either "watch out" or "be careful". Lampada is correct as it also is a way of saying goodbye. Basically, take care of yourself, something like всего хорошего, but not as formal.