Why extract DNA from certain white blood cells instead of whole blood?
There are several reasons why a lab might choose to get DNA from lymphocytes instead of whole blood. Generating genomic DNA from whole blood is not necessarily the best idea, as all the excess protein (mainly hemoglobin from RBCs) needs to be gotten rid of at some point. By narrowing down your input to just include cells that have DNA, you lessen the amount of protein you need to extract in comparison to the total amount of DNA present.

Lymphocytes include T cells, B cells, and NK cells, and are relatively easy to quickly purify, especially in a lab with the requisite experience and reagents/equipment. They are by no means the only type of white blood cells from which one could purify DNA - peripheral blood mononuclear cells (PBMCs) can also be used, for example. They contain lymphocytes as well as other cell types like monocytes, macrophages, dendritic cells, etc. (In my experience they are even more straightforward to purify from whole blood than just lymphocytes, as it is essentially a single-step protocol with a few washes afterwards.) The end result is the same, though - DNA from lymphs should be the same as DNA from PBMCs, granulocytes, or neutrophils.

Any somatic nucleated cell in the body contains "representative" DNA, as except for in a few specialized cell types like T and B cells that undergo genomic rearrangement to form the T- and B-cell receptors, respectively, all cells from an individual contain the same genomic DNA sequence. Blood is relatively easy to come by from patients and clinical study participants, which is why it is often the source of DNA for studies instead of, say, skin punches or mouth swabs.