Gozar de buena salud

Grammatically yes, but it's very rare. Usually, we say 'tener mala salud' or 'estar delicado de salud'. 'Gozar' implies enjoyment, so it's weird to 'enjoy' bad health.

Yes, it is significantly more formal and is preferred in writing and professional speech.

Yes, it is a universal Spanish expression, though its frequency in casual speech varies.

Yes, if you want to sound a bit formal or humorous about your pet's vitality.

It's a traditional grammatical structure in Spanish where certain verbs of possession or enjoyment require 'de'.

They are almost identical. 'Gozar' is slightly more traditional/literary, while 'disfrutar' is slightly more modern.

Yes, it implies the health is good by default.

Gozaba de buena salud (was healthy over time) or Gozó de buena salud (was healthy at a specific point).

Very common, especially in health, economy, and culture sections.

Only metaphorically (e.g., 'el edificio goza de buena salud' meaning it's well-maintained), but it's less common than for people or institutions.

It's an idiom for 'unbreakable health', even stronger than 'buena salud'.

¡Que goces de buena salud por muchos años!

In this context, yes. In other contexts, it just means 'to enjoy'.

Yes, you will often see it in medical certificates and reports.