I've been playing this game on and off for a long time. One thing I've yet to pick up on, however, is how purebred, themed, and exclusive sylesti work. I, for the most part, understand the themed sylesti.
How do the genetics for them work, though? I know that when a themed is bred with a non-themed, the eggling doesn't come out themed. What happens if you breed two different themed? Is there a way to breed traits into a themed sylesti's lineage?
Now, what are exclusive and purebred sylesti? Do they come with any particular perks? How do you get them? Do people value them as much as themed sylesti on the breeding and sales market?

'Themed' really only means that a pet has an official color scheme and was obtained as part of an event. Their base genetics are the exact same as any other pet. Themed pets can be listed as 'Tamed' or 'Exclusive' depending on how they're obtained. Tamed Themed pets are caught from the wild or obtained as eggs from fest zones. Themes that are labeled as Exclusive are pets that are obtained in pretty much any other way (excluding breeding). For example, the sandbox pets from this past Summer Fest. Any Themed created in the sandbox is an exclusive.

A 'Purebred' is a pet resulting from the breeding of two pets of the same theme.
For example, these 2 themed lunes (Cartha and Saturn) are both themed pets of the same theme. This pet is one of their descendants. He only has pets of the same theme in his lineage so he is considered a purebred. You can see the difference in the themed name underneath his parents names. His theme name (Sun-Kissed Dryad Lunemara) is in blue with a blue star while Saturn and Cartha's theme names are both purple with a gold star. Blue indicates a purebred pet.

If I bred 2 pets of 2 different themes together they would have a pet that did not have one of those theme names. They would not be considered purebred.

There are no perks JUST for being a themed, really, but certain ones do. Exclusive themed pets that are made in the fest growing pets minigames can be customizable to a small extent while they're growing. Themed obtained from the Lost Grove have additional stats that normal pets don't have. You can sometimes obtain high stat themed pets from other events. Mostly, themed are just valued more highly than other normal pets since they are only obtained during a certain time frame.

Purebred pricing can be difficult but in general they're worth less than a first generation themed. There are exceptions, such as purebreds that are the result of a breeding project (like max stats or 6vis).

You can breed in whatever traits you want into a purebred line so long as that trait is held by a pet of that themed already. Just breed with the pet that already has the trait and tada. It's yours now. Disruptors and Genetic Mutators can be used, of course, if the pet you want to add them to fits the requirements of the item you're using.

Did I get everything? I wrote it all kinda fast so I might have skipped some info.

Sakina's explanation is all correct except for one very small thing.

There is only one instance when breeding two different themes will result in themed offspring: breeding a Dawnbreaker aeri with a Frostkeeper aeri results in a Mistwarden aeri. Thus all Mistwardens are considered purebreds. This is the only case when cross-theme breeding can occur; in all other cases offspring of two different themes are not themed.

I knew I'd forget something. Thanks, Som.

Thank you both!