Is there an "Adam and Eve" for each new trait during (not just human) evolution?

Do new beneficial mutation evolve simultaneously and independently with several individuals of the same specie or only with one specimen who then transfers it to all of the specie?

In other words, when we talks about common ancestor we usually mean a specie, so for example, all earth dwelling reptiles, birds and mammals are the descendants of Tiktaalik (to the best of my knowledge), and we know that Tiktaalik was the first (or one of the first) species to come to land, but was there a first individual Tiktaalik who went to the land, like Tim Minchin's Tony the fish.

Imagine being the first dude ever to have feet...

Imagine you're a fish, and you are swimming in the ocean ... and suddenly out of nowhere Tony just goes "I'm getting out, I'm freezing, I'm just gonna pop out for a bit, maybe lie on my towel, I'll bring you an ice cream"

-"Don't bother ,Tony, you are a freak! Tony is a freak, let's go, let's ostracize him for being different"

It seems to me that from the same species of ancestors there are several descending distinct species, so it seems to me that that that there was a species of reptiles that at some point some individual of that group developed Dinosaur features and traits and whose descendants became Dinosaurs and then Birds, and another individual of that specie or a descending specie (not necessarily at the same time) developed mammary glands whose descendants became mammals, and other individuals who didn't mate with those two whose descendants remained reptiles.

Was there ever a first dinosaur with feathers who is the father or mother of all birds, or the first fish with lungs, the first monkey with opposable thumbs from whom all apes descend, or the first mammal that decided to go for a swim and all dolphins, whale and seals are his direct descendant of that first swimming mammal?

And also for traits that don't lead to speciation, was there a first human who was lactose tolerant, and all the lactose tolerant people on the world are his/her descendants, or the first person with no wisdom teeth, the first person with slanted eyes, and the first person with red/blond hair?

*- If there are gross inaccuracies and/or over-simplification in my understanding of the evolutionary process, this is because I'm not a biologist, and I would accept happily any correction.