Any Pokémon can have eggs with any Pokémon regardless of species or gender, and eggs can even be spawned by single or more than two Pokémon that are paired. The Pokémon just has to make a nest for the egg and wish very hard for an egg before it goes to sleep in the nest. The next morning, egg(s) have mysteriously appeared in the nest next to the Pokémon.
The Pokémon that hatch from the eggs have features from the Pokémon that wished for it, though sometimes may have different colouration from their parent(s)'.
Astrolupe is a legendary Pokémon who flies across the sky and can hear the wishes of the Pokémon wishing for eggs and flies down to use its power to give the sleeping Pokémon their eggs. No one knows where the eggs come from however, as they always appear when the Pokémon are asleep. If the Pokémon stays awake the whole night or if they have another Pokémon watching them, no eggs appear. The Pokémon assume it must be part of the ritual to be asleep and having no witnesses or it won't work.
Pokémon in this world reincarnate, though they are unaware of it. Cygnova brings the souls of deceased Pokémon into the stars where Astrolupe takes them back to bring back as eggs. Cygnova started as enthusiastic as Astrolupe, but grew bitter from seeing how negatively death is viewed, leading to him lashing out at them.
Levels exist but it takes a long time before Pokémon start to study and find out about them, as there is no way to really figure it out without considerable effort.
Pokémon can learn new moves by coming up with them naturally at certain points or by learning them from other Pokémon.
Pokémon can know more than four moves but they all tend to have four moves they prefer to use the most.
The Pokémon evolve in similar ways as normal. When they’ve reached a certain level, though unknown to themselves, using evolution items that can be found just about anywhere, by the close bond they share with others, knowing certain moves, being in certain locations, at specific times and so on. Trade evolutions are triggered by the Pokémon moving to live somewhere else.
Shinies and other non-canon colour variants are simply called colour variants and can happen fairly commonly.
Regional forms and other different forms exist and are called variant Pokémon. Their appearances are caused by them being in a different environment from their original line.
Pokémon who are a mixture of several Pokémon can evolve in all the ways their parents can, or they can choose to not evolve as well.
Legendary Pokémon have very few offspring, only rare few having more than the original. Mythical Pokémon have some offspring, they can be met sometimes and are just rare compared to regular Pokémon. The original legendary and mythical Pokémon tend to go by the names of their species, as they are the very first ones.
Legendary and mythical Pokémon cannot die of natural causes, but they can be killed. If one of the original legendary or mythical Pokémon dies, they are reincarnated again as the same Pokémon, but with no memories of their past life, being essentially a new Pokémon.