Name for interior design style- room decorated with stuffed animals, animal skins and horns

Trophy room.

Or game room, though that has other connotations also.

Big game trophy room might be appropriate.

Hunting Lodge I think would be the closest. I would say: The dining room in this castle is decorated in the style of a hunting lodge, horns covered the walls and all kinds of stuffed animals were placed around the table

Depends on the type of animals... @JJosaur's suggestion of "hunting lodge" is fine if the animals are your basic deer, fox, coyote, bear, pheasant, bobcat, etc., and your fabrics are leather and flannel check.

Skeletal remains of the heads of indigenous American fauna in a room with a bright palette, slip-covered furniture, rustic tables, (preferably fashioned from reclaimed wood) and Native American and Mexican details and accessories (throw in a cactus, or two) is "Southwestern Shabby-Chic".

Keep the skulls and cacti, change the palette to white and pale pastels, add a white leather sectional, a sheepskin rug, a large landscape painting with a "Marlboro Man" character in the distance, crystal chandeliers and cowboy decorations - you've got "Texas Millionaire".

Take the above, ditch the painting and the cowboy decorations, place an old rocking chair by a pot-bellied stove in the corner, throw a handmade quilt over the sofa, hang a flintlock rifle over the entertainment center, a stuffed jackrabbit and maybe a banjo... voila! : "Beverly Hillbillies".

If you've got lions and tigers and various heads with exotic tusks and horns, and you make use of palm trees and rattan furniture, the style is called: "Safari".

If you have exotic animals and plants with ornate Victorian furnishings in a dark palette, you've got your basic,"19th Century English Gentleman's Drawing Room."

If you mix Edwardian furniture, African and Asian animal corpses, knick-knacks from India and Chinoiserie - you'll have "British Importer".

Rustic or Frontier Style would be an appropriate description. Of course not all the elements you mention have to be present to constitute this style.

The other kind of environments that typically displays this style of decorating are the old-fashioned gentleman's club, or colonial officers' mess. Both these types might celebrate sport hunting with animal-head trophies, although it might be a stretch to say that there was an official name for this style.