Can P. falciparum and other malaria parasites be cultured and genetically engineered?
Yes it is possible to culture plasmodia but they don't grow in a simple constituted medium. Usually RPMI supplemented with serum and erythrocytes is used for growing plasmodia ex-vivo. This article discusses the issues related to plasmodial culture in detail. The authors say that sometimes a certain growth stage (in particular gametocyte) is lost on prolonged cultivation. 
In the same paper it is discussed that the exo-erythrocytic (liver stage) plasmodia can be grown on cultured hepatoma cell lines.

Several transfection methods also exist. However, culturing is not a mandatory prerequisite for transfection. Usual methods like electroporation work for a wide spectrum of cell types (in-vivo electroporation is also becoming popular). You can refer to this article for a summary on different transfection methods for plasmodia.

A major limitation, in my opinion would be that the different life stages may not be reproduced in a single culture (as you can see that each stage thrives in a very different type of milieu).