Can anyone tell me the names of some genes/mutations and what they actually look like when visible? How do I know when it's visible or just carried? Do I only get mutations if the Sylesti are inbred?
I'm asking a lot of questions but I'm really just curious about how the genetics work, and I do know how those 4-way genetic charts work if you wanna refer me to a diagram or something lol

You can play around with traits in the generator.

If you're looking at a pet, you can take a look at the genetics section. Blue is visible (e.g., AA, CC). Green is carried (e.g., NB, NQ). Purple is a visible trait with a hidden underneath (e.g. EG, KL). You can only tell if there are carries and hiddens after the pet is tested, though.

Sylestia genetics are essentially Punnet squares (the four-way genetic charts). So like if you bred pets with AG and NC in the same slot, you could get AC, CG, NA, or NG.

Mutations have nothing to do with inbreeding. Inbreeding doesn't affect anything, actually (though pets that are too closely related can't breed with each other). It's just a stat there for the curious.

One thing Somneli forgot to mention is that if you do breed two pets together with visibles, the genes will always be alphabetical in inheritance strength (not counting 'N' which is only used to signify a carry, or a blank gene essentially)

Example: You breed together two pets with genetics/mutations EG and BF
The lil pundit square will show you can get the following: EB, EF, GB, GF
But the actual way they would end up is BE, EF, BG, FG.
Which might not look like much at first, but essentially means you could never get a visible 'G' due to it being the lowest in inheritance strength. The only potential visibles are B, E, and F.

Which is why restricted traits (traits you can only get via a special themed pet, or through the use of a restricted trait philter sold in the diamond shop) are lower in the genetic strength, to make them harder to breed into a pet that doesn't already have one.

Thank you, I understand it now that was helpful. Do mutations not show up when it is a hatchling? I've noticed hatchlings say "unknown" instead of "none"

yes- hatchlings have no mutation art, and they only appear once it swaps the adolescent/adult art. If a pet has been genetested you can see what they are in the hatchling's genetics section though.