Sorry, but the answer to #1 is not correct.
1. A "List Pet" is any pet that appears on The Rares List, located here:
viewtopic.php?f=20&t=112366Although not an official list, The Rares List is a list of the rarest pets of CS, composed over many years by multiple trading experts and CS veterans. It factors both rarity and demand into play when selecting where to put a pet on the list and is a great way to determine the fairness of trades.
2. Pets are given their rarity manually, so this will usually happen after all of the pets have grown into their adult outcomes. If a pet misses getting a rarity because it hasn't finished growing at the end of the month, it will receive its rarity the following month.
3. The sitting cats were originally made to resemble the Warrior Cats in Erin Hunter's Warriors series, but had to have the names taken down due to copyright issues. Now, the lineart is used for special events instead of regular adoptables.
I hope that I could answer your questions!