I haven't had much interaction with biology since 12th grade (2010), so kindly answer in layman's terms and detail as well.
The Wikipedia page you linked says that sweat is composed of a liquid similar to blood plasma. As all DNA in humans is stored in the nucleus of a cell, it seems unlikely that the sweat itself would contain any DNA.
However, when someone sweats significantly, I can't imagine that no skin cells end up in the sweat. In any case in which this occurs, then the DNA in the cell is in the sweat. Additionally, with small amounts of sweat, I can't imagine how it would be collected without getting any skin cells in it.
In every case, what is being tested is the DNA contained in cells of human tissue
I believe the answer is no.
Sweat is the liquid secretion of body cells, not including the cells themselves. If the cells are secreted when sweating, then you may find some DNA in your sweat. You are right on that. Serum contains many different types of lymphocytes, the majority of them--mature red blood cells, don't have nucleus. So they don't contain DNA. And, live skin cells cannot be sweated. Dead skin cells are easily found on the mucosal tissues of your body, however mucosal tissues cannot sweat! In a word, it's basically impossible to see DNA in sweat.
Yes but not humans DNA
Sweat does not normally contain human DNA, it does not pick up many skin cells under normal conditions, excessive sweat can pick up some however. However sweat is colonized by bacteria fairly quickly once it reaches the surface of the skin so unless you are extracting it directly from the gland it will contain bacteria with their own DNA.