Yes, dominant and recessive genes. If you see the dominant gene (like rose comb) you can't tell whether the chicken has two genes for rose comb, or just one, because they look the same either way. If they have one gene for rose comb and one for not-rose, their chicks can inherit either one.
Yes, it is sometimes possible to work out who the parents are by paying attention to specific genes (like comb type, feathered vs. clean feet, muff/beard or not, frizzled or silkied or not, and certain colors or patterns in the feathers.) Of course there are times when you cannot figure it out for sure, because there are several possible parents that have the right genes to produce a specific chick.
I can tell that Fiona the frizzle has at least one parent with frizzled feathers. And the chicks with clean (featherless) feet & legs probably have parents with clean feet too, although that is less certain.
To the best of my knowledge, not at the present time.
I know of one company that has a few specific tests for chickens: gender, blue egg gene, and the recessive white gene (iqbirdtesting.com). But that's definitely not enough to identify specific breeds.
