As a part of my research on English language I am working on animal idioms. I would be grateful if some of you could once again answer a few questions conserning the subject:
1) Can an animal idiom be concerned as 'formal'? Are there some used typically informal, are some more used in writing (ex. brave as a lion is more used in books I guess than in a typical conversation).
2) As for such expressions as: top dog (formal? informal?), wild goose chase (formal? informal?) , queer fish, mad as a hornet, flogging a dead horse are more in use in a particular region? Do you know of any expressions that are used in the UK, but not in the US?

I know I can find most of those information in the dictionnary, but a part of the research is to make a research among the native English speakers. I will appreciate any help!
Firstly, you can't assume that

written= formal

You can get very formal speech and very informal writing.

Idioms are generally considered informal, in speech or in writing.

Some are used a lot, some are rarely used. For example your 'queer fish' sounds like something out of the 1920s to me, I've never heard anyone use it in modern days.
I agree with Nona.
"The lion's share" is commonly used and doesn't sound too informal. Same for "stir up a hornets' nest". The point is that there are degrees of formality and things that you might put in a newspaper article would obviously be out of place in a contract!