How do these characters represent the animals in the Chinese Zodiac?
Strictly speaking,「子丑寅卯辰巳午未申酉戌亥」do not represent the Chinese zodiac, but the Earthly Branches. However, there is a one-to-one mapping between the zodiac and the branches.
Which series came first?
Definitely the earthly branches, which was recorded on Shang Dynasty oracle bones - see, for example, 《甲骨文合集》37986.
There is no confirmation on how the twelve animals originated, but according to different opinions their first reference is either the Han Dynasty Records of the Grand Historian or a Tang Dynasty Buddhist manuscript 《法苑珠林》.
Is there any mapping to show how both sets of characters are related to each other?
The order which you've listed in the question is, in fact, their correspondence. To state explicitly:
Are the second string of characters based on Classical Chinese?
No. There are no confirmed origins of the 12 animals, but popular explanations say that they either come from old Chinese folk tales or Buddhist mythical animals.