Steal An Egg's official description confirms that players discover different eggs, pets, sizes and mutations, and that treadmill training grants Speed. The available sources do not establish a permanent timer, biome list or numeric Speed gate for every egg, so those details require current-server testing.
Speed and progression
The official description says that treadmill training grants Speed. It does not give biome names, gate values or a fixed progression order. Check the live gate before planning a farming route, and record the area, displayed requirement and date if you verify one in-game.
About spawn times
Do not treat a third-party “spawn time” number as universal without testing it. Server state, player activity and updates can affect what you observe. Record the biome, server conditions, observation time and whether the egg was already claimed before calling a timer confirmed.
A safer farming routine
Keep the base direction in mind, choose an egg you can return with your current Speed, and finish the delivery before chasing another spawn. A rare egg is not useful if a guard or rival player takes it before it reaches your pen.