How does the adaptive immune system store information?
There is probably no precise answer to this question since, as others pointed out, the adaptive immune system is not really comparable to a computer. That said, here are a some items you might want to look into.

Question 1: the main site of information storage for the adaptive immune system is a special region of DNA in T and B lymphocytes that encodes immunoglobulins, the proteins that these cells use to recognize foreign organisms. The sequence stored in this special DNA region can be rearranged by hypermutation and recombination mechanisms. This occurs early in the life of T and B lymphocytes, so that each cell gets a unique DNA sequence. Those cells that later on happen to recognize invaders will multiply as part of the immune response, and they will later survive as memory cells, so that their specific DNA sequences are retained. If the same infection happen again later in life, these memory cells will help the immune system respond more quickly.

Question 2: the region of DNA used is on the order of 1 million bases long, and it has been estimated that the human immune system can generate around $10^{11}$ different immunoglobulins. How to interpret these numbers in terms of "memory size" is not clear to me though. On one hand, 1 million bases is equivalent to 250 kbytes (each nucleotide base storing 2 bits); but on the other hand, $10^{11}$ as an integer can be stored in 37 bits. I guess it depends on what level of information you consider.

Question 3: there is no mechanism for directly transfering information between lymphocytes. Lymphocytes cannot read and write their DNA at will, and they are not able to simply copy sequences from each other. Rather, their diversity comes from first randomly re-shuffling their immunoglobulin-coding DNA, followed by selection of the successful cells --- this is a brute-force, random-search method, like evolution itself. Also, lymphocytes from unrelated individuals will generally interpret each other as invaders, causing an immune response. However, it is possible to experimentally "edit" the "memory" of lymphocytes extracted from blood using genetic techniques. This has for example recently been explored to "train" cancer patients own lymphocytes to attack cancer cells, in some cases with remarkably good results.