As far as I know, all Chinese restaurants prepare food to take away, especially in Madrid, Barcelona, and large cities.
About your other question, yes, they exist but there are not so many as New York City Delis, for instance. Look for establishments with "Platos preparados" signs. All El Corte Ingles stores have a prepared dishes section in their supermarkets.
Apart of these, in Barcelona (and in Madrid, too, but they are more scarce), you will find establishments whose main activity is roasting chickens and selling them. They usually prepare some side dishes too, as roasted potatoes or roasted peppers. Look for either "Pollo al ast" or "Pollo asado"(in Spanish), or "Pollastre al ast" (in Catalan) signs.