what is that DNA like sign above my pet with a number, and what does it mean?
It's the pedigree. Now what is pedigree and how is it determined, you ask? Every talent and derby ability has a rank. Every rank is worth a certain number of pedigree. Common = 1 pedigree Uncommon = 2 pedigree Rare = 3 pedigree Ultra-Rare = 4 pedigree Epic = 5 pedigree (the little pips next to each, +1, will tell you how many each is worth)
There are 10 possible talents and 10 derby abilities, for 20 total. Add up the numbers and you have the pedigree. That means the total pedigree is between 20 and 100 (I don't think either is possible).
The pedigree number really means very little in practice. Yes a high pedigree pet has more high level talents possible when you are thinking of hatching. But it says nothing about what those talents are or will be. My best pets have anywhere between 59 and 72 pedigree. I've had many failed pets with 72-80 pedigree, so it really means nothing until you train and see what you get.