If dinosaurs could have feathers, would they still be reptiles?
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This answer offers an introduction to phylogeny on the case study of dinosaurs and birds. If you are not at ease with the concept of monophyletic group, you should definitely have started with this introduction.

This post is somewhat related.

Origin of your misunderstanding

The question is all about nomenclature (and a little bit of semantics). One can call reptiles whatever (s)he wants. The question is what do we define as being a reptile? And the answer is that there are two possible definitions, a common "bad" definition (Definition 1) and a phylogenetic-based "good" definition (definition 2). I think your confusion comes from the use of the same term to mean two different (but related) things.

Definition 1: reptiles

In general, when people talk about reptiles, they talk about turtles, Rhynchocephalia, Squamata and Crocodilia. In this sense, the term reptile is NOT a monophyletic group. In other words, all species being called reptiles (according to this definition) do not share a common ancestor who has no other descendants that the species being called reptiles.

Definition 2: Reptilia

However, there is a defined monophyletic group called Reptilia. Reptilia is a group that contains all Amniota except the Synapsida (mammals and their extinct close relatives). In other words, Reptilia contains all generally called reptiles (as defined above) and all birds as well as all extinct species that are descendent from this same common ancestor.

Exploring the tree of life by yourself

You will find the tree of life on tolweb.org or on onezoom.org (see The best free and most up to date phylogenetic tree on the internet? for more info).

Using tolweb.org: here are the roots of the tree of the Amniota. And you will probably want to search for the Dinosauria (there) and see how closely related they are to the birds but not so much to the turtles.

Addressing your question directly


  If dinosaurs could have feathers, would they still be reptiles?


If by reptiles you mean Reptilia, then whether a given dinosaur has feathers, endothermy, or an exoskeleton (!) doesn't change anything to the fact that this dinosaur is a reptile.

If by reptiles you mean turtles, Rhynchocephalia, Squamata and Crocodilia, then whether a given dinosaur has feathers or not doesn't change anything to the fact that this dinosaur is NOT a reptile!

Known Dinosaurs with feathers?

Given that birds are the only animals that have feathers and that birds are dinosaurs, we don't know of any species that has a feather that is not a dinosaur! However, when we think of dinosaurs, we usually don't think about a pigeon.

Of course, the Archaeopteryx had feathers too. Archaeopteryx is an avian (bird-like) dinosaur. 



But even non-avian dinosaurs such as Velociraptor probably had feathers as well. More info about how closely related is Velociraptor to the birds (Aves) there on tolweb.org. You probably remember the so-called Velociraptor from Jurassic Park movie (on the left) but here is what a Velociraptor probably really looked like (on the right). 

 

The "velociraptors" from Jurassic park actually look more like a large Deinonychus without feathers than a Velociraptor (Thank you @Gaurav in the comments). However as Deinonychus is much harder to pronounce and way less sexy than Velociraptor, they chose to use Velociraptor in the movie Jurassic Park and deceived an entire generation about what a Velociraptor is! More information about what the species of Dinosaurs the Jurassic Park movie features in the youtube video (in French): Le Vélociraptor from Max Bird.

You might want to have a look at the post Were there any flying dinosaurs? that is very related.

At least, all these big and extinct things with scales are dinosaurs, right?

Well.... not exactly. There are species that you would probably call dinosaurs that we typically don't consider as dinosaurs. Have a look at the MinuteEarth video called "What makes a dinosaur?" for more information in this regard.

Fun and instructive videos


Les dinosaures ont disparu ? from Max Bird and Jamy (in french)
Le Vélociraptor from Max Bird (in French)
What makes a dinosaur? from MinuteEarth