I know that DNA was discovered later in time but during the 17th century, specifically around 1620 and onwards, would people have known anything about the idea of DNA? If so then how much would they have known?
In 1620 even the warped phlogiston theory was in the future; Lavoisier's and Dalton's chemistry based on atoms was in the far future. Mendel's laws of segregation and independent assortment were not even on the horizon.
To summarize, in 1620:
- They didn't know that atoms existed.
- They thought that combustion was basically a reaction of decomposition; that is, they believed that the oxides (which they called earths, hence our fossilized phrase "rare earths") were the simple substances, and burning something decomposed it into some sort of subtle hot substance and an earth.
- They thought that heredity was blending, that is, offspring blended the attributes of the parents. (As an aside, this was actually the most serious argument against Darwin's theory of natural evolution by common descent with modification; only in the 20th century, when Mendel's work was rediscovered, did biologists understand how natural selection could possibly work.)
Given the abysmal state of the entire sciences of chemistry and biology in 1620 I would say that most certainly they wouldn't even have been able to comprehend the idea that heredity was determined by a fiendishly complex chemical substance present in each and every cell of the body.
Breeding, as a concept, was understood historically quite far back in history, but not in a scientific manner. DNA as a concept would just be too foreign a thing.
It took extensively keeping records to become a thing, and if you research that you'll find that:
In Europe, the origin of animal breeding lays in the United Kingdom. It was Sir Robert Bakewell (1725 – 1795) who introduced keeping accurate records of performance of animals so that objective selection became possible.
But far back in pre-history and with the Greeks and Romans, there's evidence of selective breeding, both in crops and animals. While the concept of DNA did not exist, the concept of a family line with certain characteristics was known. And, it was understood in some places that inbreeding resulted in Bad Things. But, if it had been more properly understood, some of the things that happened in, for instance, Europe and Russia's royal line would not have happened. (Hemophilia) This is more of a history question, though I can see how it might tie into a world with similar tech.
In regards to world-building, I would say this: many people of wisdom have figured out the general shape of things, and have observed the effects of things--though not always correctly. When they have, it's often been lost to history as unproven, while wildly inaccurate things (like a mother being surprised or looking on something ugly leading to a deformed child) would often spread. The story of Jacob and the spotted sheep, when it comes down to it, is likely a story about selective breeding, but when read, it can sound like sheep looking at something and the hand of God were solely responsible. Less credulous folk would likely adhere to that interpretation.