Luck
Luck changes the chance of Epic, Legendary, Secret, Celestial and Infinity pets. A pet's chance becomes its base chance times (1 + Luck/100). With +500% Luck, a base chance of 0.005% becomes 0.03%.
Secret Chance applies after that, and only to Secret and Infinity pets: multiply again by (1 + Secret Chance/100). Celestial Chance, Infinity Chance and XL Chance work the same way for their own pets.
The Debug option in Settings shows your Luck, but the number can be wrong when an Index Luck Boost is active for a world and you hatch an egg that is not in the Index.
Shiny and Mythic
The base chance for a Shiny is 1 in 40. The base chance for a Mythic is 1 in 100. Shiny Seeker, an enchant, adds 0.2% to the Shiny chance directly. The 
Secret Elixir replaces the Mythic multiplier of the Infinity Elixir.
Other stats
Hatch Speed and Extra Hatch change how quickly eggs open and how many open at once. You can equip 4 pets by default, and Pet Equips raises that. Storage is the number of pets you can keep. Bubbles, Bubble Capacity and Bubble Sell Value are covered on the Bubbles page.