These questions can't be answered by a simple yes or no because of the way they've been asked, but it's not your fault! I'll try to explain.
To start with, DNA can be regarded as occurring mainly in the nuclei of cells (I'll leave out mtDNA here)
Going back to basics, blood is composed principally of white blood cells, red blood cells, platelets and plasma.
Now virtually all blood transfusions are transfusions of mainly red blood cells along with plasma (the liquid part of the blood). I say mainly because it's not impossible for this type of blood transfusion to contain the odd white blood cell.
Circulating red blood cells do not contain cell nuclei, chromosomes or DNA. Therefore, in normal blood transfusion you are very unlikely to donate your DNA to the recipient of your blood.
A whole blood transfusion contains white blood cells, red blood cells, platelets and plasma.
Now white blood cells do contain cell nuclei, chromosomes and DNA, so you will donate your DNA to the recipient.
However, there's really no need to be concerned for a number of reasons.
Firstly, it's simply not possible for donor DNA from white blood cells to combine with the DNA of the recipient's white blood cells and divide into daughter cells containing both types of DNA.
Secondly, white blood cells have a very unpredictable lifespan in the body which is governed by the function of the different types. Some last only hours, whilst others last years. However, because they cannot combine with the white blood cells of the recipient, they are effectively doomed as soon as they enter the bloodstream with their DNA locked away inside them.
Thirdly, whole blood transfusions are only very rarely needed in patients.
So in effect, normal blood transfusions will not give the recipient your DNA and in full blood transfusions, your