I'm not a parasitologist, and I was wondering about the current state of malaria research. For instance, is it possible to culture P. falciparum? Are there techniques for genetic manipulation, editing or transfection (analogous to in vitro cell culture) ? Are there any major limitations to studying these parasites in culture?
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).