Okay, so I was bored and waiting for some pets to finish missions and figured I'd hope over to the breeding grounds to look for potential breeding combos. But I came across it spitting out weird potential kids quite often. About every 3rd or 4th possible offspring generated, they get the seashell trait...
Example:

Potential mother: http://www.sylestia.com/view/pets/?petid=130240
Potential father: http://www.sylestia.com/view/pets/?petid=524406

They both have koi in the second gene slot...so how and why is it working out like this? Shouldn't the kids definitely have koi since it's visible in both the potential parents?

Hidden trait. There's that seashell marking behind Koi.

When you breed a Seashell with a Koi, they're both in the same slot, but both cannot show up, right?

Koi is dominant to Seashell. Thus, Koi is visible. however, Seashell is hidden behind it.
think of it like a punnet square. BBxCC (grandparents)=BC (koi vis, seashell hidden.)

Then BCxBC = 75% koi vis, 25% seashell vis.

Ahhhh, I sort of understand now. Looks like I've got some reading up to do on hidden traits then, mainly how to see if they have one and stuff, saw a guide here somewhere. :) Thanks!