Yes, CDA can be applied to the study of language (which means everyday language I guess) and literature, as for the former, there are CDA based anlayses such as about US presidential addresses, and for the latter there are anlayses about JK Rowling's Harry Potter noves.
CDA offers a way of critical thinking rather than one single path to carry out a research project as the approach does not "have a unitary theoretical framework" (Van Dijk, 2001, p. 353). In this sense, CDA is neither quantitative nor qualitative, for the approach embraces a plethora of methods and approaches which could be used in studying both language (linguistics) and literature.
Yes. I know of a junior colleague who is currently completing her PhD in African literature with a focus on HIV/AIDS in fiction and non-fiction writings and CDA is one of her methodological frameworks.
all encompassing answer would be yes it is. The discussion as to how it may be applied or what frameworks for applying CDA to the analysis of language or literature are available is a different discussion, as may be clear from the above given responses.