Is there a field in medicine called--cosmetic dermatology?

Not official. There are dermatologists who further sub-specialize by completing a fellowship in cosmetic surgery or dermatologic surgery after residency. They become more expert at performing cosmetic and surgical procedures such as resurfacing, lasers, liposuction, blepharoplasty. But there are some doctors who just say they specialize in cosmetic dermatology without proper training.
YES but not official. Dermatology has become very specialized over the last few years. A comprehensive Dermatology practice will offer services in Medical, surgical and aesthetic/cosmetic dermatology. A "cosmetic dermatologist" mainly deals with issues relating to aesthetics. So they will mainly perform skin rejuvenation/anti-aging procedures such as Botox/fillers, chemical peels and laser treatments amongst others.
Dermatology. is the study of skin(which includes but not limited to hair and nail diseases). cosmetic dermatology includes cosmetic treatment of the skin (ie fillers, neuromodulators,lasers...)
Not official. There are sub-specialties within dermatology like dermatopathology, mohs microsurgery, cosmetic dermatology, amongst others. It is a sub-specialty that even non-dermatologists can perform and it involves aesthetics like lasers, fillers, neuromodulators like botox, etc.
Yes and no.. Many dermatologists specialize in the cosmetic rather than medical aspects of skin care, because many patients come to them with problems that are largely cosmetic. A dermatologist can get fellowship training in procedural dermatology (e.g., invasive laser-based procedures, hair transplants), but the majority of dermatologists are already qualified to treat cosmetic skin disorders.
Yes. It is a subspecialty within dermatology with procedures such as facial peels and other facial rejuvenation procedures.