Dirty Dozen
Answer: OX
Samuel Yeom

Each sentence contains a reference to one of the items from a set of twelve things: The twelve months of the year, the twelve Labors of Hercules, the twelve animals in the Chinese Zodiac, and the gifts in the song "The Twelve Days of Christmas".

The flavortext tells us to find the missing number and type. The text contains three items of each type, except that there are only two animals from the Chinese Zodiac. So the missing "type" is the Chinese Zodiac. As for the number, each of the four sets of twelve have a well-defined order of its twelve items. In the text, each position is used exactly once, except that the second position is never used. So the missing "number" is two. The final answer is thus the second animal from the Chinese Zodiac.

Sentence Reference Type Number
Come and hear the story of my august accomplishments. August Months of year 8
Did you ever see a three-headed dog? Cerberus Labors of Hercules 12
Or how about a dragon? Dragon Chinese Zodiac 5
Yesterday I captured a bull just like the one that lived in a labyrinth. Cretan Bull Labors of Hercules 7
I had to do all of this with zero help - a goose egg! Geese a-laying Days of Christmas 6
On top of that, it was a long march back home. March Months of year 3
And I had to trek through all the mud - you know what they say about April showers. April Months of year 4
I tried asking the pipers I met on the way back for directions. Pipers piping Days of Christmas 11
But I couldn't hear them because a rooster was crowing at the same time. Rooster Chinese Zodiac 10
Despite all these setbacks, I must press on to my next mission of retrieving a belt from the Amazon. Belt of Hippolyta Labors of Hercules 9
The bird on that pear tree gives me hope. Partridge on a pear tree Days of Christmas 1