Alright thanks.

Second question: If one parent has a gene visible and the other one has the carried, is there a chance it would be only carried on the offspring? Or is it always visible?

Yes there is a chance the gene will only be carried. The reason for this is because a carrier has a dominant AND a recessive allele for the gene which means the offspring has the ability to either flat show the gene (pick up the dominant alleles from both parents) or become a carrier themselves (with one dominant and one recessive) allele. :3

This is excellent, thanks! But what about pairings where both parents carry the same genes and mutations? Will the offspring be guaranteed to be at least a carrier of all of those traits?

Not necessarily Apple it all depends.

If both parents Carry the genes then there is a 25% the offspring will get nothing. A 50% chance that they will carry the gene and a 25% chance that the gene will be visible.

If one of the parents has a visible gene and the other carries then there is a 50% chance that the child will be a carrier and a 50% chance that the gene will be visible.

And if both parents have it visible then it will also be visible.

So...basically expressing a trait is recessive and so the only way to get a trait expressed is to have both parents be carriers or holders of the trait?

Well don't expect a pair with only one carrier or visible,
making babies with the gene.

1 Carry : 50% will carry as well, 50% of nothing.
1 Visible : 100% Carry
(Thanks Dimitri on the fix~ ^3^")

And just like Rosedawn said :
2 Carry : 25% Visible, 50% Carry, 25% of nothing
1 Carry, 1 Visible : 50% Carry, 50% Visible

And a pair of 2 Visible will ALWAYS give their kids,
their Visible genes they both show~ owo

Hope I made it simple? owo

In fact, it is more like
1 non-carrier, 1 carrier: 50% carry, 50% have nothing
1 non-carrier, 1 visible: 100% carry

The carrier has one chance on two to pass his gene to a non-carrier while the visible always pass his gene to a non-carrier

My bad, this is what happens when I'm always tired
I can't do "math" correctly~ XD
Thanks for correcting me~ :3
It would have been troublesome if this get in the head
of some people and mess up everyone~ ^-^"

So...just to clarify, it's possible (with lots of experimenting and breeding and gens) to get a pet that displays 5 traits? (In my case, a vulnyx with visible tiger stripes, advanced runes, feathered wings, jeweled collar, and small horns)

Totally! It'll just take looooots of breeding thou~ XD
And Hopefully, you'll get a kid with all 5 Genes
showing, after all that long breeding~
(You must just be really patient and cross finger the
kids holds the right genes~ P=)