Flooring – Is a cork floor pet friendly/durable

No, not by itself. You're dealing with a spectrum of sound, cork will only partially absorb a portion of the spectrum. It will certainly improve things, depending on the existing wall surface. Also, don't forget the ceiling contributes more to reverberation than the floor.

Absorbent materials is one way to control reverberation. Another is altering the surface angles to eliminate parallel surfaces. This is why many concert halls have those odd zig-zag wall surfaces. This is also why heavy draperies are effective, besides significant absorption, what sound is reflected is done at random angles. It's certainly debatable whether these treatments are attractive or not.

You would be best served by using a variety of strategies to better address the full sound spectrum. Do not rely on one material, use a variety on every surface. Anywhere you can, even if in only a few spots, disrupt the straight walls with something with different surface angles.

Corners are also strong reflectors. Placing tall architectural or sculptural elements in corners would be beneficial. Also consider the wall/ceiling corners. Certain indirect lighting or sizable crown molding in this area will also help. While you wouldn't want a totally random hodge podge of materials for aesthetic reasons, a tasteful variety would be more attractive than every surface covered in cork, IMO. I like cork too, but not on everything! :)

You don't have to glue it but it needs to be secured somehow - duct tape. Don't want the pieces rolling up on each other over time.

You would just be duct taping the underlayment to the outside flooring or in the inside seams. And duct tape isn't hillbillying your floor. It is used because how well it handles moisture. Maybe someone has a specific underlayment tape they like to use but several manufacturers (I read directions) said to use duct tape.

Also try to get some duct tape attached to the subfloor for each piece - clean subfloor. So around the perimeter you are taping your cork to the subfloor directly along the outside edge. In the "middle" you want to tape cork piece #1 to the subfloor. Then you tape some of cork piece #2 to the subfloor. Then you add additional tape to tape #1 and #2 together, where ever they join. This is not more work either. If you only attach the pieces together at the seams they will slide on you.

(anyone who edits this to give a better picture if free to do so)

So your choices are glue everything down. Or tape the seams to each other and subfloor. I have always had the choice to do both and always chose tape. Not sure when I would glue - generally this would scare me because lets say you spill some lemonade on the floor and it gets to the subfloor. The acidity could eat through the glue (duct tape wouldnt be affected). Yes I understand that it would have to be a lot of lemonade to make a difference but if it happens right at a seam and two pieces lost their glueing...